Counting Principles

What you will learn: Solve simple counting problems Use the Fundamental Counting Principle to solve counting problems Use permutations to solve counting problems Use combinations to solve counting problems

Fundamental Counting Principle The Fundamental Counting Principle states that if on e event can occur m ways and a second event can o ccur n ways, the number of ways the two events can occur in sequence is m n. Ex. How many different pairs of letters from the English alphabet are possible? – There are 2 events in this situation. The first event is the choice of 1st letter, and the second is the choice of the second letter. – 26 x 26 = 676

E2 Telephone numbers in the US currently have 10 digits. The first three are the area code and the next seven are the local telephone number. How many different telephone numbers are possible within each area code? (Note at this time, a local telephone number cannot begin with 0 or 1)

E2 Answer Because the first digit of a local telephone number cannot be 0 or 1 there are only 8 choices for the first digit. For each of the other six digits there are 10 choices. 8 x 10 x 10 x 10 x 10 x 10 x 10 = 8,000,000

Permutations A permutation is an ordered arrangement of n different elements. A permutation of n different elements is an ordering of the elements such that one element is first, one is second, one is third, and so on.

E3 How many permutations are possible for the letters A, B, C, D, E, and F? Answer – First position: 6 possible – 2 nd Position: 5 possible – 3 rd Position: 4 possible – 4 th : 3 possible – 5 th : 2 possible – 6: 1 letter remaing – 6!=6 x 5 x 4 x 3 x 2 x 1 – =720

Number of Permutations of n Elements n (n—1) =n! In other words there are n! different ways that n elements can be ordered

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.9 Formula for n P r The formula for the number of permutations of n elements taken r at a time is # in the collection # taken from the collection

E4 Eight horses are running a race. In how many different ways can these horses come in 1 st, 2 nd, and 3 rd ? (No ties) Answer: – 1 st : 8 possibilities – 2 nd :7 Possibilities – 3 rd : 6 possibilities – = 336

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.11 The T’s are not distinguishable. STATSSTATS Example: Distinguishable Permutation If some of the items are identical, distinguishable permutations must be used. In how many distinguishable ways can the letters STATS be written? STATS The S’s are not distinguishable. Example continues.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.12 Definition of Distinguishable Permutation The number of distinguishable permutations of the n objects is The letters STATS can be written in S’s T’s A’s where n = n 1 + n 2 + n n k.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.13 Definition of Combination A combination of n elements taken r at a time is a subset of the collection of elements where order is not important. Using the letters A, B, C, and D, find all the possible combinations using two of the letters. {AB} {AC} {AD} {BC} {BD} {CD} This is the same as {BA}. There are six different combinations using 2 of the 4 letters.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.14 Fornula for n C r The formula for the number of combinations of n elements taken r at a time is # in the collection # taken from the collection

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.15 Example: Combination Example 6: How many different ways are there to choose 6 out of 10 books if the order does not matter? There are 210 ways to choose the 6 books. 3

E7 How many distinguishable ways can the letters BANANA be written? Answer: There are 6 letters 3 are A’s 2 are N’s 1 is a B

E8 You are forming a 12-member swim team for 10 girls and 15 boys. The team must consist of 5 girls and 7 boys. How many 12-member teams are possible?

E8 There are ₁₀C₅ ways of choosing 5 girls. There are ₁₅C₇ ways of choosing 7 boys. By the FCP, there are ₁₀C₅ ₁₅C₇ ways of choosing 5 girls and 7 boys. 1,621,610