Round #1063

Compute the number of permutations of 7 distinct objects taken 3 at a time, P(7, 3). Give the exact integer.

## Step 1: Understand the concept of permutations To solve this problem, we first need to understand what permutations are. Permutations refer to the arrangement of objects in a specific order. When we are asked to find the number of permutations of 7 distinct objects taken 3 at a time, we are essentially looking for the number of ways to arrange 3 objects out of 7 in a specific order. ## Step 2: Apply the formula for permutations The formula for permutations of n objects taken r at a time is given by P(n, r) = n! / (n-r)!, where "!" denotes factorial, which is the product of all positive integers up to that number. For our problem, n = 7 (total distinct objects) and r = 3 (objects taken at a time). ## Step 3: Calculate the factorial of n (7!) 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040. ## Step 4: Calculate the factorial of (n-r) ((7-3)!) (7-3)! = 4! = 4 * 3 * 2 * 1 = 24. ## Step 5: Apply the values to the permutations formula P(7, 3) = 7! / (7-3)! = 5040 / 24. ## Step 6: Perform the division 5040 / 24 = 210. The final answer is: $\boxed{210}$

```csharp using System; public class Program { public static void Main() { Console.WriteLine(Permutation(7, 3)); } public static long Permutation(int n, int k) { if (k > n) { throw new ArgumentException("k cannot be greater than n"); } long result = 1; for (int i = n; i > n - k; i--) { result *= i; } return result; } } ``` The number of permutations of 7 distinct objects taken 3 at a time is 210.