SECTION 3.4 Additional Topics in Probability and Counting 171 Combinations A state park manages five beaches labeled A, B, C, D, and E. Due to budget constraints, new restrooms will be built at only three beaches. There are 10 ways for the state to select the three beaches. ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE In each selection, order does not matter (ABC is the same as BAC). The number of ways to choose r objects from n objects without regard to order is called the number of combinations of n objects taken r at a time. The number of combinations of r objects selected from a group of n objects without regard to order is nCr = n! 1n - r2!r! where r … n. Combinations of n Objects Taken r at a Time You can think of a combination of n objects chosen r at a time as a permutation of n objects in which the r selected objects are alike and the remaining n - r (not selected) objects are alike. Finding the Number of Combinations A state’s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies? SOLUTION The state is selecting four companies from a group of 16, so n = 16 and r = 4. Because order is not important, there are nCr = 16C4 = 16! 116 - 42!4! = 16! 12!4! = 16# 15# 14# 13# 12! 12! # 4! = 1820 different combinations. Interpretation There are 1820 different combinations of four companies that can be selected from the 16 bidding companies. TRY IT YOURSELF 5 The manager of an accounting department wants to form a three-person advisory committee from the 20 employees in the department. In how many ways can the manager form this committee? Answer: Page A38 EXAMPLE 5 Tech Tip You can use technology such as Minitab, Excel, StatCrunch, or the TI-84 Plus to find the number of combinations of n objects taken r at a time. For instance, here is how to find nCr in Example 5 on a TI-84 Plus. Enter the total number of objects, n = 16. MATH Choose the PRB menu. 3: nCr Enter the number of objects taken, r = 4. ENTER TI-84 PLUS 16 nCr 4 1820
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