Section 11.7 The Binomial Theorem Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Objectives Expand a power of a binomial using Pascal’s triangle or factorial notation. Find a specific term of a binomial expansion. Find the total number of subsets of a set of n objects.

Pascal’s Triangle

The Binomial Theorem Using Pascal’s Triangle

Example Expand (u  v)4. Solution: We have (a + b)n, where a = u, b = v, and n = 4. We use the 5th row of Pascal’s Triangle: 1 4 6 4 1 Then we have:

Example Expand (x  3y)4. Solution: Here a = x, b = 3y, and n = 4. We use the 5th row of Pascal’s triangle: 1 4 6 4 1 Then we have

The Binomial Theorem Using Combination Notation

Example

Finding a Specific Term Finding the (k + 1)st Term The (k + 1)st term of (a + b)n is

Example Find the 7th term in the expansion (x2  2y)11. Solution: First, we note that 7 = 6 + 1. Thus, k = 6, a = x2, b = 2y, and n = 11. Then the 7th term of the expansion is

Total Number of Subsets The total number of subsets of a set with n elements is 2n.

Example The set {A, B, C, D, E, F} has how many subsets? Solution: The set has 6 elements, so the number of subsets is 26 or 64.