Combinatorics: Combinations Section 11.6 Combinatorics: Combinations Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Objectives Evaluate combination notation and solve related applied problems.

Combinations We sometimes make a selection from a set without regard to order. Such a selection is called a combination.

Example Find all the combinations of 2 letters taken from a set of 3 letters {A, B, C}. Solution: The combinations are {A, B}, {A, C}, and {B, C}. There are 3 combinations of the 3 letters taken 2 at a time.

Definitions Subset: Set A is a subset of set B, denoted A  B, if every element of A is an element of B. Combination: A combination containing k objects is a subset containing k objects. Combination Notation: The number of combinations of n objects taken k at a time is denoted nCk.

Combinations of n Objects Taken k at a Time The total number of combinations of n objects taken k at a time, denoted nCk , is given by

Binomial Coefficient Notation Another kind of notation for nCk is binomial coefficient notation.

Example Evaluate , using forms (1) and (2). Solution: a) By form (1), b) By form (2), The 8 tells where to start. The 3 tells how many factors there are in both the numerator and the denominator and where to start in the denominator.

Subsets of Size k and of Size n  k The number of subsets of size k of a set with n objects is the same as the number of subsets of size n  k. The number of combinations of n objects taken k at a time is the same as the number of combinations of n objects taken n  k at a time.

Example A team manager has 11 students who are qualified to play basketball. How many different 5-person teams can be chosen? Solution: There are 462 different 5-person teams that could be chosen.

Example How many committees can be formed from a group of 8 seniors and 10 juniors if each committee consists of 5 seniors and 6 juniors? Solution: The 5 seniors can be selected in 8C5 ways and the 6 juniors can be selected in 10C6 ways. If we use the fundamental counting principle, it follows that the number of possible committees is