# 6.8 - Pascal's Triangle and the Binomial Theorem 11.1 Pascals Triangle and the Binomial Theorem FACTORIALS, COMBINATIONS Factorial is denoted by the symbol !. The factorial of a number is calculated by multiplying all integers from the number to 1. Formal Definition The symbol n!, is define as the product of all the integers from n to 1. In other words, n! = n(n - 1)(n 2)(n 3) 3 2 1 Also note that by definition, 0! = 1 Example #9

3! 3 2 1 6 (9 3)! 6! 6 5 4 3 2 1 720 9! 9 8 7 6 5 4 3 2 1 362,880 Combinations Definition Combinations give the number of ways x element can be selected from n distinct elements. The total number of combinations is given by, n Cx and is read as the number of combinations of n elements selected x at a time.

The formula for the number of combinations for selecting x from n distinct elements is, Note: n! n Cx x !(n x )! n! n! n! 1

n Cn n !(n n)! n ! 0! n ! n C0 n! n! n! 1 0!( n 0)! 0! n ! n ! Combinations Example #10

2 5! 5! 5 4 3 2 1 10 5 C3 3!(5 3)! 3! 2! 3 2 1 2 1 7! 7! 7 6 5 4 3 2 1

35 7 C4 4!(7 4)! 4! 3! 4 3 2 1 3 2 1 4 C0 1 3 C3 1 The Binomial Theorem Strategy only: how do we expand these?

1. 3. (x + 2)2 (x 3)3 2. 4. (2x + 3)2 (a + b)4 The Binomial Theorem Solutions

1. (x + 2)2 = x2 + 2(2)x + 22 = x2 + 4x + 4 2. (2x + 3)2 = (2x)2 + 2(3)(2x) + 32 = 4x2 + 12x + 9 3. (x 3)3 = (x 3)(x 3)2 = (x 3)(x2 2(3)x + 32) = (x 3)(x2 6x + 9) = x(x2 6x + 9) 3(x2 6x + 9) = x3 6x2 + 9x 3x2 + 18x 27 = x3 9x2 + 27x 27 4. (a + b)4 = (a + b)2(a + b)2 = (a2 + 2ab + b2)(a2 + 2ab + b2) = a2(a2 + 2ab + b2) + 2ab(a2 + 2ab + b2) + b2(a2 + 2ab + b2) = a4 + 2a3b + a2b2 + 2a3b + 4a2b2 + 2ab3 + a2b2 + 2ab3 + b4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 THAT is a LOT of work! Isnt there an easier way?

Introducing: Pascals Triangle Row 5 Row 6 Take a moment to copy the first 6 rows. What patterns do you see?

The Binomial Theorem Use Pascals Triangle to expand (a + b)5. Use the row that has 5 as its second number. The exponents for a begin with 5 and decrease. 1a5b0 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1a0b5 The exponents for b begin with 0 and increase. In its simplest form, the expansion is a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5. Row 5 The Binomial Theorem Use Pascals Triangle to expand (x 3)4.

First write the pattern for raising a binomial to the fourth power. 1 4 6 4 1 Coefficients from Pascals Triangle.

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 Since (x 3)4 = (x + (3))4, substitute x for a and 3 for b. (x + (3))4 = x4 + 4x3(3) + 6x2(3)2 + 4x(3)3 + (3)4 = x4 12x3 + 54x2 108x + 81 The expansion of (x 3)4 is x4 12x3 + 54x2 108x + 81. The Binomial Theorem For any positive integer, n (a b) n a n n! n! n!

n! a n 1b a n 2b 2 a n 3b 3 ... a n r b r ... b n 1!(n 1)! 2!(n 2)! 3!(n 3)! r!(n r )! n! n Cx x !(n x )!

The Binomial Theorem Use the Binomial Theorem to expand (x y)9. Write the pattern for raising a binomial to the ninth power. (a + b)9 = 9C0a9 + 9C1a8b + 9C2a7b2 + 9C3a6b3 + 9C4a5b4 + 9C5a4b5 + 9C6a3b6 + 9C7a2b7 + 9C8ab8 + 9C9b9 Substitute x for a and y for b. Evaluate each combination. (x y)9 = 9C0x9 + 9C1x8(y) + 9C2x7(y)2 + 9C3x6(y)3 + 9C4x5(y)4 + 9C5x4(y)5 + 9C6x3(y)6 + 9C7x2(y)7 + 9C8x(y)8 + 9C9(y)9 = x9 9x8y + 36x7y2 84x6y3 + 126x5y4 126x4y5 + 84x3y6 36x2y7 + 9xy8 y9 The expansion of (x y)9 is x9 9x8y + 36x7y2 84x6y3 + 126x5y4 126x4y5 + 84x3y6 36x2y7 + 9xy8 y9.

Lets Try Some Expand the following a) (x-y5)3 b) (3x-2y)4 Lets Try Some Expand the following (x-y5)3 Lets Try Some Expand the following

(3x-2y)4 Lets Try Some Expand the following (3x-2y)4 How does this relate to probability? You can use the Binomial Theorem to solve probability problems. If an event has a probability of success p and a probability of failure q, each term in the expansion of (p + q)n represents a probability.

Example: 10C2 * p8 q2 represents the probability of 8 successes in 10 tries The Binomial Theorem Brianna makes about 90% of the shots on goal she attempts. Find the probability that Bri makes exactly 7 out of 12 consecutive goals. Since you want 7 successes (and 5 failures), use the term p7q5. This term has the coefficient 12C5. Probability (7 out of 10) = 12C5 p7q5 12! = 5! 7! (0.9)7(0.1)5

The probability p of success = 90%, or 0.9. = 0.0037881114 Simplify. Bri has about a 0.4% chance of making exactly 7 out of 12 consecutive goals.