pencil, red pen, highlighter, GP notebook, calculator U11D11 Have out: Bellwork: Expand and simplify (a + 4b)3. 1 Start with row 3 of Pascal’s triangle. +1 1 1 Write powers of a in descending order. +1 1 2 1 +1 Write powers of 4b in ascending order. 1 3 3 1 1 a3 (4b)0 + 3 a2 (4b)1 + 3 a1 (4b)2 + 1 a0 (4b)3 a3 + 12a2b + 48ab2 + 64b3 +1 +1 +1 +1 total:

Yesterday we used Pascal’s Triangle to find the coefficients of the terms of a binomial expansion. Example: (a + b)4 1 2 3 4 6 Start with row 4 of Pascal’s triangle. Write powers of a in descending order. Write powers of b in ascending order. 1 a4 b0 + 4 a3 b1 + 6 a2 b2 + 4 a1 b3 + 1 a0 b4 a4 + 4a3b + 6a2b2 + 4ab3 + b4

The coefficients can also be rewritten using combinations. Use the notation ____________. (a + b)4 = a4 b0 + a3 b1 + a2 b2 + a1 b3 + a0 b4 Notice some patterns: The sum of the powers of “a” and “b” equal the binomial’s power. The “r” and the power of “b” are always equal This pattern holds true for the expansion of any binomial ______. (a + b)n

Binomial Theorem For every positive integer n, (a + b)n = n n 1 n 2 n n 1 n 2 n n – 2 n n – 1 n Example: Use the Binomial Theorem to write the expansion of (x + y)5. x5 y0 + x4 y1 + x3 y2 + x2 y3 + x1 y4 + x0 y5 x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5

Under the expansion, label: 1st term, 2nd term, 3rd term, etc. + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 1st term 2nd term 3rd term 4th term 5th term 6th term n + 1 Notice that the number of terms in this expression is ______. one more For any binomial, the number of terms is always ___________ than the binomial’s ______. power Examples: How many terms are in the expansions for the following binomials? a) (3x + y)9 b) (2x – y)17 c) (x + 4y)125 10 terms 18 terms 126 terms

The Binomial Theorem is also very useful whenever we need to find specific terms in an expansion. To find the rth term of the binomial expansion (a + b)n, use the formula: Examples: Find the specified term of each binomial expansion. a) Fourth term of (x + y)5. x5 – 4 + 1 y4 – 1 x2 y3 10x2y3

To find the rth term of the binomial expansion (a + b)n, use the formula: Examples: Find the specified term of each binomial expansion. b) Third term of (x – y)20. x20 – 3 + 1 (–y)3 – 1 Shortcuts: Notice that these are always equal. x18 (–y)2 The powers of x and (-y) must add to 20. So, the power of x must be 20 – 2 = 18 190x18y2

Practice: Find the specified term of each binomial expansion. a) Fifth term of (x + y)12 b) Second term of (x – y)10 x12 – 5 + 1 y5 – 1 x10 – 2 + 1 (–y)2 – 1 x8 y4 x9 (–y)1 495x8y4 –10x9y c) Fourth term of (x + 2)8 d) Third term of (2x + 1)6 x8 – 4 + 1 (2)4 – 1 (2x)6 – 3 + 1 (1)3 – 1 x5 (2)3 (2x)4 (1)2 56 • x5 • 8 = 448x5 15 • 16x4 • 1 = 240x4

Finish the worksheet And IC - 125, 126, 131

 # of ways to select 4 heads from 8 tosses 1. Use Pascal’s Triangle and/or combinations to determine the following probabilities. a) P (4 heads in 8 tosses)  # of ways to select 4 heads from 8 tosses  total # of possible outcomes in 8 tosses 0H 1H 2H 3H 4H 5H 6H 7H 8H row 8 

 # of ways to select 2 heads from 6 tosses 1. Use Pascal’s Triangle and/or combinations to determine the following probabilities. b) P (2 heads in 6 tosses)  # of ways to select 2 heads from 6 tosses  total # of possible outcomes in 6 tosses 0H 1H 2H 3H 4H 5H 6H row 6 

 # of ways to select 6 heads from 9 tosses 1. Use Pascal’s Triangle and/or combinations to determine the following probabilities. c) P (6 heads in 9 tosses)  # of ways to select 6 heads from 9 tosses  total # of possible outcomes in 9 tosses 0H 1H 2H 3H 4H 5H 6H 7H 8H 9H row 9 

d) P (at most 2 heads in 5 tosses) 1. Use Pascal’s Triangle and/or combinations to determine the following probabilities. d) P (at most 2 heads in 5 tosses) + + 0H 1H 2H 3H 4H 5H row 5 

e) P (at least 5 heads in 7 tosses) 1. Use Pascal’s Triangle and/or combinations to determine the following probabilities. e) P (at least 5 heads in 7 tosses) + + 0H 1H 2H 3H 4H 5H 6H 7H row 7 

f) P (odd number of heads in 6 tosses) 1. Use Pascal’s Triangle and/or combinations to determine the following probabilities. f) P (odd number of heads in 6 tosses) + + 0H 1H 2H 3H 4H 5H 6H row 6 

2. Determine the probabilities for the following 5–card poker hands using a standard deck of cards. a) all red # of possible 5–card hands of that are all red. P (all red) = Total # of possible 5–card hands b) all clubs # of possible 5–card hands of that are all clubs. P (all clubs) = Total # of possible 5–card hands

2. Determine the probabilities for the following 5–card poker hands using a standard deck of cards. c) 3 diamonds and 2 spades # of ways to get 3 cards from all diamonds # of ways to get 2 cards from all spades P (3♦, 2♠) = Total # of possible 5–card hands

2. Determine the probabilities for the following 5–card poker hands using a standard deck of cards. f) 3 two’s, 1 ten, 1 other Think about the groups you are choosing from. There are: four 2’s, four 10’s, 44 cards that are not 2’s or 10’s # of ways to get 3 cards from all 2’s # of ways to choose 1 card from all 10’s # of ways to choose 1 card from everything else Total # of possible 5–card hands