Given a stick of size N, find the number of ways in which it can be cut into K pieces such that length of every piece is greater than 0.
Input : N = 5 K = 2 Output : 4 Input : N = 15 K = 5 Output : 1001
Solving this question is equivalent to solving the mathematics equation x1 + x2 + ….. + xK = N
We can solve this by using the bars and stars method in Combinatorics, from which we obtain the fact that the number of positive integral solutions to this equation is (N – 1)C(K – 1), where NCK is N! / ((N – K) ! * (K!)), where ! stands for factorial.
In C++ and Java, for large values of factorials, there might be overflow errors. In that case we can introduce a large prime number such as 107 + 7 to mod the answer. We can calculate nCr % p by using Lucas Theorem.
However, python can handle large values without overflow.
Extend the above problem with 0 length pieces allowed. Hint : The number of solutions can similarly be found by writing each xi as yi – 1, and we get an equation y1 + y2 + ….. + yK = N + K. The number of solutions to this equation is (N + K – 1)C(K – 1)
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