Sec. 12.8 Counting Problems

1
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

2
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

3
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

4
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

5
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

6
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

7
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

8
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

9
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
3
3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

10
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
3
2
3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

11
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
3
2
1
3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

12
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

13
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

14
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

15
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

16
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

17
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

18
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

19
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

20
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

21
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2
1

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

22
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2
1
6!

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

23
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2
1
6!
720

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

24
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2
1
6!
720

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

___ ___ ___
25
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2
1
6!
720

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

___ ___ ___
5
26
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2
1
6!
720

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

___ ___ ___
5
4
27
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2
1
6!
720

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

___ ___ ___
5
4
3
28
Sec. 12.8 Counting Problems

• How many different telephone numbers can a County
  use if a phone number
• is to begin with 207 followed by four
  digits?

207- ___ ___ ___ ___
10
10
10
10
10,000

• How many distinct ways can the letters x, y, z
  be arranged if repetition of
• letters is not permitted?

___ ___ ___
New Notation
3
2
1
3!
6
3. In how many ways can six different books be
arranged on a shelf?
___ ___ ___ ___ ___ ___
6
5
4
3
2
1
6!
720

• Consider the five letters a, b, c, d, e. In how
  many distinct ways can three letters
• be selected and arranged if repetition is
  not allowed?

___ ___ ___
5
4
3
60
29
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
30
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
31
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
32
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
33
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
34
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
35
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
36
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
37
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
38
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
39
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
40
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
41
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
___ ___ ___ ___ ___
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
42
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
___ ___ ___ ___ ___
5
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
43
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
___ ___ ___ ___ ___
5
4
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
44
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
___ ___ ___ ___ ___
5
4
3
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
45
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
___ ___ ___ ___ ___
5
4
3
2
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
46
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
___ ___ ___ ___ ___
5
4
3
2
1
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
47
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
___ ___ ___ ___ ___
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
48
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
___ ___ ___ ___ ___
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
49
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
5!
___ ___ ___ ___ ___
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
50
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
5!
___
___ ___ ___ ___ ___
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
51
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
5!
___
___ ___ ___ ___ ___
2!
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
52
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
5!
___
___ ___ ___ ___ ___
3!
2!
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
53
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

___ ___ ___ ___ ___
3!
2!
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
54
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
55
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
9. In how many different ways can the letters of
the word success be arranged?
56
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
6!
___
9. In how many different ways can the letters of
the word success be arranged?
57
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
6!
___
3!
9. In how many different ways can the letters of
the word success be arranged?
58
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
6!
___
2!
3!
9. In how many different ways can the letters of
the word success be arranged?
59
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

2!
3!
9. In how many different ways can the letters of
the word success be arranged?
60
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

2!
3!
12
9. In how many different ways can the letters of
the word success be arranged?
61
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

60
2!
3!
12
9. In how many different ways can the letters of
the word success be arranged?
62
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

60
2!
3!
12
9. In how many different ways can the letters of
the word success be arranged?
7!
___
63
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

60
2!
3!
12
9. In how many different ways can the letters of
the word success be arranged?
7!
___
3!
64
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

60
2!
3!
12
9. In how many different ways can the letters of
the word success be arranged?
7!
___
2!
3!
65
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

60
2!
3!
12
9. In how many different ways can the letters of
the word success be arranged?
5040
7!
___
_____

2!
3!
66
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

60
2!
3!
12
9. In how many different ways can the letters of
the word success be arranged?
5040
7!
___
_____

2!
3!
12
67
Sec. 12.8 Counting Problems

• A club consisting of seven people must choose a
  president, vice-president, and
• a secretary. How many different
  arrangements or permutations are possible?

___ ___ ___
7
6
5
210

• Go-cart world has eight cars available. If three
  people must select a car in which
• to ride, how many different permutations of
  car and driver are possible?

___ ___ ___
8
7
6
336
7. In how many different ways can the letters a,
a, c, c, c be arranged?
a b c d e
a a c c c
120
5!
___
___

10
___ ___ ___ ___ ___
3!
2!
12
5
4
3
2
1
120
8. In how many different ways can the letters of
the word banana be arranged?
720
6!
___
___

60
2!
3!
12
9. In how many different ways can the letters of
the word success be arranged?
5040
7!
___
_____

420
2!
3!
12