Space and time efficient Binomial Coefficient

Write a function that takes two parameters n and k and returns the value of Binomial Coefficient C(n, k). For example, your function should return 6 for n = 4 and k = 2, and it should return 10 for n = 5 and k = 2.

We have discussed a O(n*k) time and O(k) extra space algorithm in this post. The value of C(n, k) can be calculated in O(k) time and O(1) extra space.

C(n, k) = n! / (n-k)! * k!
        = [n * (n-1) *....* 1]  / [ ( (n-k) * (n-k-1) * .... * 1) * 
                                    ( k * (k-1) * .... * 1 ) ]
After simplifying, we get
C(n, k) = [n * (n-1) * .... * (n-k+1)] / [k * (k-1) * .... * 1]

Also, C(n, k) = C(n, n-k)  // we can change r to n-r if r > n-r 

Following implementation uses above formula to calculate C(n, k)

C/C++

// Program to calculate C(n ,k)
#include <stdio.h>

// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff(int n, int k)
{
    int res = 1;

    // Since C(n, k) = C(n, n-k)
    if ( k > n - k )
        k = n - k;

    // Calculate value of [n * (n-1) *---* (n-k+1)] / [k * (k-1) *----* 1]
    for (int i = 0; i < k; ++i)
    {
        res *= (n - i);
        res /= (i + 1);
    }

    return res;
}

/* Drier program to test above function*/
int main()
{
    int n = 8, k = 2;
    printf ("Value of C(%d, %d) is %d ", n, k, binomialCoeff(n, k) );
    return 0;
}

Java

// Program to calculate C(n ,k) in java
class BinomialCoefficient
{
    // Returns value of Binomial Coefficient C(n, k)
    static int binomialCoeff(int n, int k)
    {
    	int res = 1;
    
    	// Since C(n, k) = C(n, n-k)
    	if ( k > n - k )
    		k = n - k;
    
    	// Calculate value of [n * (n-1) *---* (n-k+1)] / [k * (k-1) *----* 1]
    	for (int i = 0; i < k; ++i)
    	{
    	res *= (n - i);
    	res /= (i + 1);
    	}
    
    	return res;
    }
    
    /* Driver program to test above function*/
    public static void main(String[] args)
    {
    	int n = 8;
    	int k = 2;
    	System.out.println("Value of C("+ n + ", " + k+ ") "
    							+ "is" + " "+ binomialCoeff(n, k));
    }

}
// This Code is Contributed by Saket Kumar

Python

# Python program to calculate C(n ,k)

# Returns value of Binomial Coefficient
# C(n, k)
def binomialCoefficient(n, k):
    # since C(n, k) = C(n, n - k)
    if(k > n - k):
        k = n - k
    # initialize result
    res = 1
    # Calculate value of 
    # [n * (n-1) *---* (n-k + 1)] / [k * (k-1) *----* 1]
    for i in range(k):
        res = res * (n - i)
        res = res / (i + 1)
    return res

# Driver program to test above function 
n = 8
k = 2
res = binomialCoefficient(n, k)
print("Value of C(%d, %d) is %d" %(n, k, res))

# This code is contributed by Aditi Sharma
Value of C(8, 2) is 28

Time Complexity: O(k)
Auxiliary Space: O(1)

This article is compiled by Aashish Barnwal and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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