Counting Techniques

1
Counting Techniques

• Worst case.
• Ordered places or for each.
• 3. Permutation (Can be done by number 2).
• 4. Combination.

2
Worst Case

• Of the science books in a certain supply room,
  50 are on botany, 65 are on zoology, 90 are on
  physics. 50 are on geology, and 110 are on
  chemistry. If science books are removed randomly
  from the supply room, how many must be removed to
  ensure that 80 of the books removed are on the
  same science?
•  
•   (A) 81  (B) 159  (C) 166  (D) 285 
  (E) 324

3
Ordered Places 1

• Katie must place five stuffed animals--a
  duck, a goose, a panda, a turtle and a swan in a
  row in the display window of a toy store. How
  many different displays can she make if the duck
  and the goose must be either first or last?
•  
• (A) 120 (B) 60 (C) 24 (D) 12 (E)
   6

4
Ordered Places 2

• The president of a country and 4 other
  dignitaries are scheduled to sit in a row on the
  5 chairs represented above. If the president must
  sit in the center chair, how many different
  seating arrangements are possible for the 5
  people?
•  
•  

 (A) 4  (B) 5  (C) 20  (D) 24  (E) 120
5
Ordered Places 3

• In how many arrangements can a teacher seat 3
  girls and 3 boys in a row of 6 if the boys are to
  have the first, third, and fifth seats?
•  
• (A)      6
• (B)     9
• (C)     12
• (D)     36
• (E)    720

6
FOR EACH

• If a customer makes exactly 1 selection from
  each of the 5 categories listed below, what is
  the greatest number of different ice cream
  sundaes that a customer can create? 
•  
• 12 ice cream flavours
• 10 kinds of candy
•   8 liquid toppings
•   5 kinds of nuts
•   With or without whip cream.
•  
• (A) 9600 (B) 4800 (C) 2400 (D) 800 (E) 400

7
Ordered Places or Permutation

• Given a selected committee of 8, in how many
  ways, can the members of the committee divide the
  responsibilities of a president, vice president,
  and secretary?
• (A) 120 (B) 336 (C) 56 (D) 1500 (E) 100

8
Ordered Places or Permutation

•  How many four-digit numbers can you form using
  ten numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) if
  the numbers can be used only once?
• (A) 72 (B) 5000 (C) 4536 (D) 10 000 (E) 210

9
Combination 1

• From a group of 8 secretaries, select 3
  persons for a promotion. How many distinct
  selections are there?
• (A) 56 (B) 336 (C) 512 (D) 9 (E) 200

10
Combination 2
A person has the following bills 1, 5, 10,
20, 50. How many unique sums can one form using
any number of these bills only once? (A) 10
(B) 15 (C) 31 (D) 35 (E) 40
11
Combination 3 and For each
From a class consisting of 12 computer science
majors, 10 mathematics majors, and 9 statistics
majors, a committee of 4 computer science majors,
4 mathematics majors, and 3 statistics majors is
to be formed. How many distinct committees are
there? (A) 1000 (B) 5270 (C) 9 327 (D) 12 345
(E) 8 731 800
12
Combination 4 and For each
A three-person committee must be chosen from a
group of 7 professors and 10 graduate students.
If at least one of the people on the committee
must be a professor, how many different groups of
people could be chosen for the committee?   (A) 
70 (B)  560 (C) 630 (D) 1,260 (E) 1,980
13
Combination 5 and For each
There are 11 top managers that need to form a
decision group. How many ways are there to form a
group of 5 if the President and Vice President
are not to serve on the same team? (A) 200 (B)
315 (C) 378 (D) 425 (E) 498