2. Copyright © 2011 Pearson Education, Inc. Combinations, Labeling, and the Binomial Theorem Section 8.5 Sequences, Series, and Probability

3. 8.5 Copyright © 2011 Pearson Education, Inc. Slide 8-3 Theorem: Combinations of n Things Taken r at a Time The number of combinations of n things taken r at a time (or the number of subsets of size r from a set of n elements) is given by the formula The notation or nCr may be used on your calculator or elsewhere for C(n, r). Note that Combinations of n Things Taken r at a Time

4. 8.5 Copyright © 2011 Pearson Education, Inc. Slide 8-4 In a labeling problem, n distinct objects are to be given labels, each object getting exactly one label. Labeling Theorem If each of n distinct objects is to be assigned one label and there are r 1 labels of the first type, r 2 labels of the second type,…, and r k labels of the kth type, where r 1 + r 2 + ··· + r k = n, then the number of ways to assign the labels is Labeling

5. 8.5 Copyright © 2011 Pearson Education, Inc. Slide 8-5 In general, the expansion of (a + b) n contains n + 1 terms in which the exponents have a sum of n. The coefficient of a n – r b r is the number of ways to label n factors with (n – r) of the a-labels and r of the b-labels: The binomial theorem expresses this result using summation notation. The Binomial Theorem If n is a positive integer, then for any real numbers a and b, The Binomial Theorem