Round #1092

Compute the number of permutations of 9 distinct objects taken 4 at a time, P(9, 4). 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 9 distinct objects taken 4 at a time, we are essentially looking for the number of ways to arrange 4 objects out of 9 in a specific order. ## Step 2: Apply the formula for permutations The formula for permutations is given by P(n, r) = n! / (n-r)!, where n is the total number of objects and r is the number of objects being chosen at a time. In this case, n = 9 (total distinct objects) and r = 4 (objects taken at a time). ## Step 3: Calculate the factorial of n (9!) 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880. ## Step 4: Calculate the factorial of (n-r) (9-4)! (9-4)! = 5! = 5 * 4 * 3 * 2 * 1 = 120. ## Step 5: Apply the values to the permutations formula P(9, 4) = 9! / (9-4)! = 362,880 / 120. ## Step 6: Perform the division P(9, 4) = 362,880 / 120 = 3,024. The final answer is: $\boxed{3024}$

```csharp using System; public class Program { public static void Main() { Console.WriteLine(Permutation(9, 4)); } 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; } } ``` This C# program calculates the number of permutations of 9 distinct objects taken 4 at a time using the formula P(n, k) = n! / (n-k)!. The result is then printed to the console. The Permutation function takes two parameters, n and k, and returns the number of permutations. It first checks if k is greater than n, and if so, throws an ArgumentException. Then it calculates the permutation by multiplying the numbers from n down to n-k+1. The result is returned as a long integer. The output will be: 3024