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

2. Objectives
Evaluate combination notation and solve related applied problems.

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

4. 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.

5. 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.

6. 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

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

8. 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.

9. 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.

10. 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.

11. 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