Given N and K. The task is to count the number of the integral solutions of a linear equation having N variable as given below:
x1 + x2+ x3…+ xN-1+…+xN = K
Input: N = 3, K = 3 Output: 10 Input: N = 2, K = 2 Output: 3
Approach: This problem can be solved using the concept of Permutation and Combination. Below are the direct formulas for finding non-negative and positive integral solutions respectively.
Number of non-negative integral solutions of equation x1 + x2 + …… + xn = k is given by (n+k-1)! / (n-1)!*k!.
Number of positive integral solutions of equation x1 + x2 + ….. + xn = k is given by (k-1)! / (n-1)! * (k-n)!.
Below is the implementation of above approach:
Applications of the above concepts:
- Number of non-negative integral solutions of equation x1 + x2 +…+ xn = k is equal to the number of ways in which k identical balls can be distributed into N unique boxes.
- Number of positive integral solutions of equation x1 + x2 + … + xn = k is equal to the number of ways in which k identical balls can be distributed into N unique boxes such that each box must contain at-least 1 ball.
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