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Chapter 11: Problem 4

Does the problem involve permutations or combinations? Explain your answer. (It is not necessary to solve the problem.) Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500\), in how many different ways can the prizes be awarded?

Short Answer

Expert verified
There are \(C(50, 3) = \frac{50!}{3!(50 - 3)!}\) ways to award the prizes.

Step by step solution

01

Understanding the Problem

The problem is about choosing 3 winners out of 50 raffle ticket purchasers. Each ticket holder has an equal probability of winning, and there is no preferential ranking among the 3 winners (that is, the first chosen is not ranked higher than the second or third.) As such, the order of selection does not matter in this case, which leads to the use of combinations instead of permutations.
02

Using the Formula for Combinations

The combination formula is represented as \(C(n, k) = \frac{n!}{k!(n - k)!}\) where \(n\) is the total number of items, \(k\) is the number of items to choose, and \( ! \) denotes a factorial, meaning the product of all positive integers up to that number. The symbol \(C(n, k)\) represents the number of combinations of \(n\) items taken \(k\) at a time.
03

Calculating the Combinations

Substitute the number of raffle ticket purchasers (n = 50) and the number of winners (k = 3) into the combination formula. This yields \( C(50, 3) = \frac{50!}{3!(50 - 3)!} \).

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