Problem 20 - College Algebra

Question

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?

Step-by-Step Solution

Answer

The problem involves combinations, and the number of ways the prizes can be awarded is given by \( C(50, 3) = \frac{50!}{3!(50 - 3)!} \)

Step 1: Understand the problem

The problem asks for the total number of ways the prizes can be awarded among 50 raffle ticket purchasers. This is a combinations problem because the order of selection doesn't matter.

Step 2: Apply the formula for combinations

The formula for combinations is given by \( C(n, r) = \frac{n!}{r!(n - r)!} \), where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) denotes factorial. For this problem, \( n = 50 \) and \( r = 3 \)

Step 3: Calculate the number of combinations

Plug the given values into the combination formula. So, \( C(50, 3) = \frac{50!}{3!(50 - 3)!} \). This will give the total number of ways the prizes can be awarded.

Key Concepts

Permutations vs Combinations

When tackling problems in algebra involving the arrangement or selection of items, it's essential to understand the difference between permutations and combinations, as they each quantify different scenarios.