23. Footrace There are 72 runners in a 10-kilometer race. How many ways can the runners finish first, second, and third?

Verified step by step guidance

1. Step 1: Recognize that this is a permutation problem because the order in which the runners finish (first, second, and third) matters.
2. Step 2: Use the formula for permutations, which is P(n, r) = n! / (n - r)!, where n is the total number of items (runners) and r is the number of positions to fill (first, second, and third).
3. Step 3: Substitute the given values into the formula: n = 72 (total runners) and r = 3 (positions to fill). The formula becomes P(72, 3) = 72! / (72 - 3)!.
4. Step 4: Simplify the denominator: (72 - 3)! = 69!, so the formula becomes P(72, 3) = 72 × 71 × 70 (since the factorial terms beyond 69! cancel out).
5. Step 5: Multiply the remaining terms (72 × 71 × 70) to find the total number of ways the runners can finish first, second, and third.

Key Concepts

Permutations

Permutations refer to the different arrangements of a set of items where the order matters. In this context, we are interested in the arrangements of the top three finishers among 72 runners. The formula for permutations is given by n! / (n - r)!, where n is the total number of items, and r is the number of items to arrange.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is essential for solving problems involving arrangements and selections.

Combinatorial Counting

Combinatorial counting involves determining the number of ways to choose or arrange items from a larger set. In this problem, we need to count the specific arrangements of the top three finishers from the total of 72 runners. This concept is crucial for solving problems in probability and statistics, especially when dealing with competitions or selections.