1. Section 10-3 • Using Permutations and Combinations

2. Using Permutations and Combinations • Permutations • Combinations • Guidelines on Which Method to Use

3. Permutations In the context of counting problems, arrangements are often called permutations; the number of permutations of n things taken r at a time is denoted nPr. Applying the fundamental counting principle to arrangements of this type gives nPr= n(n – 1)(n – 2)…[n – (r – 1)].

4. Factorial Formula for Permutations The number of permutations, or arrangements, of n distinct things taken r at a time, where rn, can be calculated as

5. Example: Permutations Evaluate each permutation. a) 5P3 b) 6P6 Solution

6. Example: IDs How many ways can you select two letters followed by three digits for an ID if repeats are not allowed? Solution There are two parts: 1. Determine the set of two letters. 2. Determine the set of three digits. Part 1 Part 2

7. Example: Building Numbers From a Set of Digits How many four-digit numbers can be written using the numbers from the set {1, 3, 5, 7, 9} if repetitions are not allowed? Solution Repetitions are not allowed and order is important, so we use permutations:

8. Combinations In the context of counting problems, subsets, where order of elements makes no difference, are often called combinations; the number of combinations of n things taken r at a time is denoted nCr.

9. Factorial Formula for Combinations The number of combinations, or subsets, of n distinct things taken r at a time, where rn, can be calculated as Note:

10. Example: Combinations Evaluate each combination. a) 5C3 b) 6C6 Solution

11. Example: Finding the Number of Subsets Find the number of different subsets of size 3 in the set {m, a, t, h, r, o, c, k, s}. Solution A subset of size 3 must have 3 distinct elements, so repetitions are not allowed. Order is not important.

12. Example: Finding the Number of Subsets A common form of poker involves hands (sets) of five cards each, dealt from a deck consisting of 52 different cards. How many different 5-card hands are possible? Solution Repetitions are not allowed and order is not important.

13. Guidelines on Which Method to Use