In C language, what is the maximum value of a binomial coefficient?

Given a positive integer ‘N’. We need to find the largest coefficient term among all binomial coefficients.

The binomial coefficient sequence is _nC₀, _nC₁, _nC₂,…,_nC_r,…,_nC_n-2,_n C_n-1, _nC_n

Find the maximum value of _nC_r.

<sub>n</sub>C<sub>r</sub> = n! / r! * (n - r)!

Input - N=4

Output - Maximum coefficient - 6

Explanation - ₄C₀= 1, ₄C₁ = 4, ₄C₂ = 6 , ₄C₃ = 4, ₄C₄ = 1

Therefore, in this case, the maximum coefficient is 6.

Input - N=5

Output - Maximum coefficient - 10

Explanation - ₅C₀= 1, ₅C₁ = 5, ₅C₂ =10, ₅C₃ = 10, ₅C₄ = 5, ₅C₅ = 1

So, in this case, the maximum coefficient is 10.

The method used in the following program is as follows

• We get the input of N from the user.

• The function maxCoeff(int n) accepts one argument 'n' and returns the largest coefficient found so far in C[n 1][n 1].

• Initialize the min and max variables with 0. 'min' is used to iterate over the C[][] array, and 'max' is used to store the maximum coefficient value found.

• Use a loop of i from 0 to n to initialize the C[][] array.

• Now traverse to the smaller of 'i' or 'n' in another loop.

• If i==j, then C[i][j]==1. Otherwise, C[i][j] = C[i-1][j-1] C[i-1][j].

• Now iterate through the entire C[][] again and store the maximum coefficient in max.

• Return results.

Example

Demonstration

#include <stdio.h>
int maxCoeff(int n){
   int C[n+1][n+1];
   int max=0,min=0;
   // Calculate value of Binomial Coefficient in
   for (int i = 0; i <= n; i++){
      min=i<n?i:n;
      for (int j = 0; j <= min; j++){
         if (j == 0 || j == i)
            C[i][j] = 1;
         else
            C[i][j] = C[i-1][j-1] + C[i-1][j];
      }
   }
   for (int i = 0; i <= n; i++){
      max = max> C[n][i] ? max: C[n][i];
   }
   return max;
}
int main(){
   int N = 3;
   printf("Maximum Coefficient :%d", maxCoeff(N) );
   return 0;
}

Output

If we run the above code, the following output will be generated −

Maximum Coefficient: 3

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