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Chapter 11: Problem 46

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 19,600 different ways to award the prizes.

Step by step solution

01

Understanding the problem Scope

The problem mentions there are 50 people (this sets our total number of different elements, which we'll denote as 'n') and we are selecting 3 winners (this is the amount we'll pick at each instance denoted as 'r'). From that, we can understand the problem's scope.
02

Apply Combination Formula

As per the problem, the order of the winners is not important. Thus, we'll use a combination. The formula for combination is given by \( C(n, r) = n! / r!(n - r)! \), where '!' indicates factorial, 'n' is the total number of items, and 'r' is the number of items to choose. In our case, 'n' is 50 (number of people bought the ticket) and 'r' is 3 (number of winning tickets). Substituting these values in the formula, we get \(C(n, r) = C(50, 3) = 50! / 3!(50 - 3)!\).
03

Simplify the Result

On solving the factorial expression, we get the total number of different ways to award the prizes. Use the properties of factorials to simplify the above equation. Thus, we get \(C(50, 3) = 19600\).

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