Write a function that takes two parameters n and k and returns the value of Binomial Coefficient C(n, k).
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!) = 20
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) // r can be changed to n-r if r > n-r
- 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.
Following implementation uses above formula to calculate C(n, k).
Value of C(8, 2) is 28
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 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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