Space and time efficient Binomial Coefficient
Here the function takes two parameters n and k and returns the value of Binomial Coefficient C(n, k).
Example:
Input: n = 4 and k = 2 Output: 6 Explanation: 4 C 2 is 4!/(2!*2!) = 6
Input: n = 5 and k = 2 Output: 10 Explanation: 5 C 2 is 5!/(3!*2!) = 10
We have discussed 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.
Approach:
- Change r to n-r if r is greater than n-r. and create a variable to store the answer.
- Run a loop from 0 to r-1
- In every iteration update ans as (ans*(n-i))/(i+1) where i is the loop counter.
- So the answer will be equal to ((n/1)*((n-1)/2)*…*((n-r+1)/r) which is equal to nCr.
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) // r can be changed to n-r if r > n-r
Following implementation uses the above formula to calculate C(n, k).
C++
// Program to calculate C(n, k) #include <bits/stdc++.h> using namespace std; // 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; } // Driver Code int main() { int n = 8, k = 2; cout << "Value of C(" << n << ", " << k << ") is " << binomialCoeff(n, k); return 0; } // This is code is contributed by rathbhupendra |
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; } /* Driver 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 |
Python3
# 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 |
C#
// C# Program to calculate C(n, k) using System; 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 Code public static void Main() { int n = 8; int k = 2; Console.Write( "Value of C(" + n + ", " + k + ") " + "is" + " " + binomialCoeff(n, k)); } } // This Code is Contributed by // Smitha Dinesh Semwal. |
PHP
<?php // Program to calculate C(n, k) // Returns value of Binomial // Coefficient C(n, k) function binomialCoeff( $n , $k ) { $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 ( $i = 0; $i < $k ; ++ $i ) { $res *= ( $n - $i ); $res /= ( $i + 1); } return $res ; } // Driver Code $n = 8; $k = 2; echo " Value of C ($n, $k) is " , binomialCoeff( $n , $k ); // This code is contributed by ajit. ?> |
Javascript
<script> // Program to calculate C(n, k) // Returns value of Binomial Coefficient C(n, k) function binomialCoeff(n, k) { let 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 (let i = 0; i < k; ++i) { res *= (n - i); res /= (i + 1); } return res; } // Driver Code let n = 8; let k = 2; document.write( "Value of C(" + n + ", " + k + ") " + "is" + " " + binomialCoeff(n, k)); </script> |
Output
Value of C(8, 2) is 28
Complexity Analysis:
Time Complexity: O(r) A loop has to be run from 0 to r. So, the time complexity is O(r).
Auxiliary Space: O(1) As no extra space is required.
This article is compiled by Aashish Barnwal and reviewed by the w3wiki team.
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