Warmup

1
Warm-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
2
10. P(inappropriate) combinations of 2
inappropriate songs
combinations that 2 songs can be played
3
11. P(2 hearts) combinations
of 2 hearts
combinations that 2 cards from a deck of 52
4
12. P(2 aces) combinations of
2 aces
combinations that 2 cards from a deck of 52
5
13. P(3 males,1 female) combs. of 3
males and 1 female
combs. for a committee
of 4 from 10
6
14. 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.

8
California 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
9
California 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
10
The 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

11
The 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?
12
Example Find the 9th term of (x y)12

• 12! 4!8!
• 12 11 10 9 8! 4 3
  2 1 8!

11880 24 495x4y8
13
Example 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
14
Example 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!
15
6.8 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

16
Warm-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