Problem 60 - Algebra and Trigonometry

Question

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000\), second prize is \(\$ 500,\) and third prize is \(\$ 100,\) in how many different ways can the prizes be awarded?

Step-by-Step Solution

Answer

The prizes can be awarded in 117,600 different ways.

Step 1: Understand permutations

The concept of permutation is used when the arrangement or order of things matter. There are three prizes which will be awarded to three different people. Therefore, in this situation, who gets the first prize, who gets the second, and who gets the third is important as each prize is of a different value.

Step 2: Calculate total number of possibilities for the first prize

There are 50 raffle tickets in total. Any one of them can be chosen for the first prize. So, there are 50 different ways to award the first prize.

Step 3: Calculate total number of possibilities for the second prize

For the second prize, as one ticket has already been selected for the first prize, only 49 tickets are left in consideration. So, there are 49 different ways to award the second prize.

Step 4: Calculate total number of possibilities for the third prize

Similarly, for the third prize, as two tickets have already been selected for the other prizes, 48 tickets are left. So, there are 48 different ways to award the third prize.

Step 5: Calculate total permutations

To find the total number of different ways the prizes can be awarded, multiply the number of ways of awarding the first prize, the second prize, and the third prize. Using the multiplication principle of counting, the expression becomes \(50 \times 49 \times 48\).