Maximum binomial coefficient term value in C

In C, finding the maximum binomial coefficient term for a given positive integer 'N' involves calculating all binomial coefficients _nC_r and determining the largest value among them.

The binomial coefficient series is _nC₀, _nC₁, _nC₂, ..., _nC_r, ..., _nC_n-2, _nC_n-1, _nC_n.

Syntax

nCr = n! / (r! * (n - r)!)

For N=4: ₄C₀=1, ₄C₁=4, ₄C₂=6, ₄C₃=4, ₄C₄=1. Maximum coefficient is 6.

For N=5: ₅C₀=1, ₅C₁=5, ₅C₂=10, ₅C₃=10, ₅C₄=5, ₅C₅=1. Maximum coefficient is 10.

Method 1: Using Pascal's Triangle Approach

This method builds Pascal's triangle using dynamic programming to calculate binomial coefficients efficiently −

#include <stdio.h>

int maxCoeff(int n) {
    int C[n+1][n+1];
    int max = 0;
    
    // Build Pascal's triangle using dynamic programming
    for (int i = 0; i <= n; i++) {
        for (int j = 0; j <= i; j++) {
            if (j == 0 || j == i) {
                C[i][j] = 1;
            } else {
                C[i][j] = C[i-1][j-1] + C[i-1][j];
            }
        }
    }
    
    // Find maximum coefficient in nth row
    for (int j = 0; j <= n; j++) {
        if (C[n][j] > max) {
            max = C[n][j];
        }
    }
    
    return max;
}

int main() {
    int N = 5;
    printf("For N = %d:<br>", N);
    printf("Maximum Coefficient: %d<br>", maxCoeff(N));
    
    N = 4;
    printf("For N = %d:<br>", N);
    printf("Maximum Coefficient: %d<br>", maxCoeff(N));
    
    return 0;
}

For N = 5:
Maximum Coefficient: 10
For N = 4:
Maximum Coefficient: 6

Method 2: Mathematical Property Approach

The maximum binomial coefficient is always _nC_n/2 (for even n) or _nC_(n-1)/2 and _nC_(n+1)/2 (for odd n) −

#include <stdio.h>

long factorial(int n) {
    if (n == 0 || n == 1) return 1;
    return n * factorial(n - 1);
}

long binomialCoeff(int n, int r) {
    return factorial(n) / (factorial(r) * factorial(n - r));
}

int maxCoeffOptimized(int n) {
    int r = n / 2;  // Middle position gives maximum coefficient
    return binomialCoeff(n, r);
}

int main() {
    int N = 6;
    printf("For N = %d:<br>", N);
    printf("Maximum Coefficient: %d<br>", maxCoeffOptimized(N));
    
    N = 7;
    printf("For N = %d:<br>", N);
    printf("Maximum Coefficient: %d<br>", maxCoeffOptimized(N));
    
    return 0;
}

For N = 6:
Maximum Coefficient: 20
For N = 7:
Maximum Coefficient: 35

Key Points

• The maximum binomial coefficient occurs at the middle of the binomial expansion.
• Pascal's triangle method has O(n²) time complexity but builds all coefficients.
• Mathematical property method has O(n) time complexity for direct calculation.
• For large values of n, factorial calculations may overflow − consider using iterative multiplication.

Conclusion

Finding the maximum binomial coefficient can be done efficiently using the mathematical property that it occurs at the middle position. The Pascal's triangle approach is useful when all coefficients are needed, while the direct calculation is optimal for finding just the maximum value.