Example: Pascal's Triangle using nCr in Java

// Pascal's Triangle using nCr
import java.util.Scanner;

// Main class containing the entry point of the program
public class Main {

    // Function to calculate nCr (n choose r)
    // C(n, r) = n! / (r! * (n-r)!)
    public static long nCr(int n, int r) {
        if (r < 0 || r > n) {
            return 0; // Invalid combination
        }
        if (r == 0 || r == n) {
            return 1;
        }
        if (r > n / 2) { // Optimization: C(n, r) = C(n, n-r)
            r = n - r;
        }
        long res = 1;
        for (int i = 0; i < r; i++) {
            res = res * (n - i) / (i + 1);
        }
        return res;
    }

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        System.out.print("Enter the number of rows for Pascal's Triangle: ");
        int numRows = scanner.nextInt();

        // Step 1: Iterate through each row
        for (int i = 0; i < numRows; i++) {
            // Step 2: Add leading spaces for proper triangle alignment
            for (int space = 0; space < numRows - 1 - i; space++) {
                System.out.print("  "); // Two spaces for wider alignment
            }

            // Step 3: Iterate through each element in the current row
            for (int j = 0; j <= i; j++) {
                System.out.printf("%4d", nCr(i, j)); // Calculate and print nCr
            }
            System.out.println(); // Move to the next line after each row
        }
        scanner.close();
    }
}