Notes 26 Combination Problems Combinations A combination of

Notes #26 Combination Problems

Combinations A combination of items occurs when • The items are selected from the same group. • No item is used more than once. • The order of items makes no difference. 9/25/2021 Section 11. 3 2

Permutation vs Combination Permutation problems: 1) Involve situations in which order matters. Ex. Ranking, seating order, positions Combination problems 1) involve situations in which the order of items makes no difference. Ex. Putting members in a Committee

Distinguishing Between Permutations and Combinations Is it a Permutation or Combination? Problem #1: • 6 students are running for student government President, Vice-President and Treasurer. ANS: Permutations, b/c order matters since we are talking about three different positions. 9/25/2021 Section 11. 3 4

Is it a Permutation or Combination? Problem #2 • 6 people are on the board of supervisors for your neighborhood park. • A three-person committee is needed to study the possibility of expanding the park. • How many different committees could be formed from the six people? ANS: Combinations, b/c the order in which the three people are selected does not matter since they are not filling different roles. 9/25/2021 Section 11. 3 5

A Formula for Combinations The number of possible combinations if “r” items are taken from “n” items is: 9/25/2021 Section 11. 3 6

Using the Formula for Combinations EX 1: How many three-person committees could be formed from 8 people? Set Up (3 Steps) 1) List what is Given: Selecting 3 people (r = 3) from 8 people (n = 8) 2) Apply the Combination Formula: 3) Show Work: FINAL ANS: 56 three-person committees could be formed from 8 people. 9/25/2021 7

Using the Formula for Combinations and the Fundamental Counting Principle EX 2: The U. S Senate of the 104 th Congress consisted of 54 Republicans and 46 Democrats. How many committees can be formed if each committee must have 3 Republicans (from 54 members) and 2 Democrats (from 46)? Q 1: Is this a permutation or combination problem? ANS: Combination, since the order in which members are selected does not matter. 9/25/2021 8

First, select 3 Republicans out of 54 Republicans: Set Up (3 Steps) 1) Let n = 54 (Total Republicans); r = 3 representative (from 54) 1) Apply Combination Formula: 3) SHOW WORK 9/25/2021 Section 11. 3 9

Secondly, select 2 Democrats out of 46 Democrats: Set Up 1) Let n = 46 (Total Democrats); r = 2 representative (from 46) 1) Apply Combination Formula: 3) SHOW WORK 9/25/2021 Section 11. 3 10

How many committees can be formed if each committee must have 3 Republicans (from 54 members) and 2 Democrats (from 46)? Set Up: Use the Fundamental Counting Principle to find the number of committees that can be formed. Republican Party: Democratic Party: Multiply the combinations 24, 804 x 1, 035 FINAL ANS: 25, 675, 140 committees can be formed if each committee must have 3 Republicans & 2 Democrats.