1. Section 10-3
Using Permutations and Combinations

2. Using Permutations and 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 r n, 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 r n, 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
Permutations
Combinations
Number of ways of selecting r items out of n items
Repetitions are not allowed
Order is important.
Order is not important.
Arrangements of n items taken r at a time
Subsets of n items taken r at a time
nPr = n!/(n – r)!
nCr = n!/[ r!(n – r)!]
Clue words: arrangement, schedule, order
Clue words: group, sample, selection