Question

There are 25 runners competing in the 100 meter dash. The top 3 runners will be awarded the following medals: 1st - gold; 2nd - silver; and 3rd - bronze. How many ways can the medals be awarded?

15625
72
13800
75

Answer 1

To find the number of ways the medals can be awarded to the top 3 runners from 25 competitors, we need to consider that we are selecting and ordering 3 runners from a total of 25.

The first step is to select the runner who will receive the gold medal (1st place), then the runner who will receive the silver medal (2nd place), and finally the runner who will receive the bronze medal (3rd place). This can be thought of as a permutation of selecting 3 runners from 25.

The number of ways to choose and order 3 runners from a group of 25 can be calculated using the formula for permutations:

\[
P(n, r) = \frac{n!}{(n-r)!}
\]

Where:
- \( n \) is the total number of runners (25).
- \( r \) is the number of runners to select (3).

Applying the values:

\[
P(25, 3) = \frac{25!}{(25-3)!} = \frac{25!}{22!}
\]

This simplifies to:

\[
P(25, 3) = 25 \times 24 \times 23
\]

Now we compute this step-by-step:

1. \( 25 \times 24 = 600 \)
2. \( 600 \times 23 = 13800 \)

Thus, the number of ways to award the medals to the top 3 runners is:

\[
\boxed{13800}
\]