Permutations and Combinations in Counting Problems
Learn about permutations and combinations and how to apply them in counting problems with clear examples and guidelines on method selection.
Permutations and Combinations in Counting Problems
E N D
Presentation Transcript
Section 10-3 • Using Permutations and Combinations
Using Permutations and Combinations • Permutations • Combinations • Guidelines on Which Method to Use
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)].
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
Example: Permutations Evaluate each permutation. a) 5P3 b) 6P6 Solution
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
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:
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.
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:
Example: Combinations Evaluate each combination. a) 5C3 b) 6C6 Solution
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.
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.
