University:
Santa Ana College
Course:
MATH 084 | Beginning and Intermediate Algebra
Academic year:

2023
Views:

438

Pages:

2
Author:

Clint W.

Combinations Subsets and Combinations Subsets: Set B is a subset of A if each member of B is also a member of A. The empty set {} is considered to be a subset of every set. Combinations of n Objects Taken r at a Time: C(n, r) = n! / (n-r)! * r! EXAMPLE 1 - Determining Subsets a. List all two-letter subsets of the set {J, K, L, M}. {J,K}, {J,L}, {J,M}, {K,L}, {K,M}, {L,M} b. List all three-letter subsets of the set {J, K, L, M}. {J,K,L}, {J,K,M}, {J,L,M}, {K,L,M} EXAMPLE 2 - Using Combinations a. A baseball game starts with 9 players. How many ways can you select 2 co-captains? 9C2 = 36 b. Chris, Sarah, Caitlin, and Peter have volunteered to help a non-profit organization build houses. Three of them will be assigned to work as painters. How many choices are there for the three painters? 4C3 = 4 c. How many combinations of the letters in the word ABSOLUTE are there, taking the letters three at a time? 8C3 = 56 d. Parents have 10 books that they can read to their children. Five are nonfiction and five are fiction. How many different sets of 4 books can they choose? 10C4 = 210 e. In how many groups of 4 books are all of the books either nonfiction or fiction? 5C4 + 5C4 = 5 + 5 = 10 Multiple Events Example 2e was a multiple event because... Two different options! Multiply Combinations: AND Add Combinations: OR EXAMPLE 3 - Deciding When to Add or Multiply Combinations For a-b, the student senate consists of 6 seniors, 5 juniors, 4 sophomores, and 3 freshmen. a. How many different committees of exactly 2 seniors and 2 juniors can be chosen? 6C2 * 5C2 = 15 * 10 = 150 b. How many different committees of at most 4 students can be chosen? 18C4 + 18C3 + 18C2 + 18C1 = 3060 + 816 + 153 + 18 = 4047 c. In how many ways can a student select 6 courses from 7 electives and 5 core courses if 3 electives and 3 core courses must be taken? 7C3 * 5C3 = 35 * 10 = 350 EXAMPLE 4 - Combinations with Cards a. From a standard deck of cards, how many hands of 5 cards all from the same suit can be dealt? 4 * 13C5 = 5148 b. How many 5-card hands having exactly 3 diamonds, 2 clubs, and 2 other cards can be dealt? 13C3 * 13C2 * 26C2 = 286 * 78 * 325 = 7,250,100 Find the number of possible 5-card hands that contain the cards specified. The cards are from a standard 52-card deck. a. 5 red cards 26C5 = 65,780 b. 4 tens and 1 jack 4C4 * 4C1 = 4 c. 2 face cards and 3 non-face cards 12C2 * 40C3 = 66 * 9880 = 652,080 d. 5 clubs or 5 red cards 13C5 + 26C5 = 1287 + 65780 = 67,067 e. At most one heart 13C1 * 39C4 + 13C0 * 39C5 = 11,157,578 f. At least 3 queens 4C3 * 48C2 + 4C4 * 48C1 = 1128 + 48 = 1176

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