Round #1061

Compute the binomial coefficient C(8, 3). Give the exact integer.

## Step 1: Understand the binomial coefficient formula The binomial coefficient, often referred to as "n choose k", can be calculated using the formula: C(n, k) = n! / [k!(n-k)!], where "n!" denotes the factorial of n, which is the product of all positive integers up to n. ## Step 2: Apply the formula to the given problem For C(8, 3), we will use the formula with n = 8 and k = 3. So, C(8, 3) = 8! / [3!(8-3)!]. ## Step 3: Calculate the factorials - 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320 - 3! = 3 * 2 * 1 = 6 - (8-3)! = 5! = 5 * 4 * 3 * 2 * 1 = 120 ## Step 4: Plug the factorials into the formula C(8, 3) = 40320 / (6 * 120) ## Step 5: Perform the calculation C(8, 3) = 40320 / 720 = 56 The final answer is: $\boxed{56}$

```csharp public class BinomialCoefficientCalculator { public static int Calculate(int n, int k) { if (k > n - k) { k = n - k; } int result = 1; for (int i = 0; i < k; i++) { result = result * (n - i) / (i + 1); } return result; } public static void Main(string[] args) { Console.WriteLine(Calculate(8, 3)); } } ```