Title: Warmup
1Warm-up
- For the Virginia State Lottery, you must choose 6
numbers out of 44 possible choices. How many
combinations of choices are there? - If you purchase 10 tickets, what is the
probability that you will win the Virginia State
Lottery?
44C6 44!
38!6! 7,059,052
P(winning) 10
7,059,052 .000001417
210. P(inappropriate) combinations of 2
inappropriate songs
combinations that 2 songs can be played
311. P(2 hearts) combinations
of 2 hearts
combinations that 2 cards from a deck of 52
412. P(2 aces) combinations of
2 aces
combinations that 2 cards from a deck of 52
513. P(3 males,1 female) combs. of 3
males and 1 female
combs. for a committee
of 4 from 10
614. P(1 blue,1 red) combs. of 1 blue
and 1 red
combs. of 2 marbles from 10
7- a) 10 b) 1 c) 315
d) 9 - 100C2 4950
- 9C3 84
- 52C2 1326
- 10C2 45
- 4C1 7C4 3C1 4C2 2520
- 12C3 20C5 3410880
- 8C5 56
- 9. 12.
- 10. 13.
- 11. 14.
8California Lottery
- Super LOTTO Plus
- Select 5 numbers from 1 47 select the MEGA
Number from 1 27.
47C5 27C1
1,533,939 27 41,416,353
9California Lottery
- MEGA MILLIONS
- Select 5 numbers from 1 56 select the MEGA
Number from 1 46.
56C5 46C1
3,819,816 46 175,711,536
10The Binomial Theorem
- Recall that a binomial has two terms...
- (x y)
- The Binomial Theorem gives us a method to expand
binomials raised to powers such as - (x y)4 (x y)5 (x y)8 (2a 3y)8
11The Binomial Theorem
- The binomial expansion of (x y)n is
- (x y)n xn nxn1 y n!
xn-m ym nxyn1 yn
(n m)!m! - The coefficient xn my m is denoted by
Example. Find the binomial coefficients for
Here are the coefficients for (x y)6
6! 4!2! 6 5 4! 4!
2 1
15
This is the coefficient for the 3rd term of (x
y)6 What about the variables?
12Example Find the 9th term of (x y)12
- 12! 4!8!
- 12 11 10 9 8! 4 3
2 1 8!
11880 24 495x4y8
13Example Find the 7th term of (3a 2b)11
- 11! 5!6!
- 11 10 9 8 7 6! 5 4 3 2
1 6!
Use
462x5y6 Replace x with (3a) and replace y
with (2b) 462(3a)5(2b)6 462(243a5)(64b6)
7185024a5b6
14Example Expand (x 3)5
- Here are the coefficients
1 5 10 10 5
1 which translates to x5 5x4y 10x3y2
10x2y3 5xy4 y5 Replace the ys with 3...
x5 5x4(3) 10x3(3)2 10x2(3)3 5x(3)4
(3)5 Simplify... x5 15x4 90x3 270x2
405x 243
Summary The Binomial Theorem is a method to
find coefficients when expanding a binomial. It
is used mainly to find a particular term.
n! (n m)!m!
156.8 Binomial Theorem Part I
- 1. x8 40x7 700x6
- 2. 32x 1
- 3. 192192x6y8
- 24x7
- 5. x8 16x7 112x6 448x5 1120x4 1792x3
1792x2 1024x 256 - 6. 64x6 192x5y 240x4y2 160x3 y3 60x2 y4
12xy5 y6 - 7. 16x4 32x3 24x2 8x 1
16Warm-up
- Tell whether each of the following is a
combination or a permutation - Nine books placed in a row on a shelf.
- Three books selected from a collection of 20
books. - An arrangement of the letters in the word BOOK.
- In how many ways can a committee of 6 be chosen
from 5 teachers and 4 students if the committee
must include 3 teachers and 3 students?
Permutation Combination
Permutation
40 different ways