GATE | GATE-CS-2003 | Question 34

m identical balls are to be placed in n distinct bags. You are given that m ≥ kn, where, k is a natural number ≥ 1. In how many ways can the balls be placed in the bags if each bag must contain at least k balls?

(A) A
(B) B
(C) C
(D) D

Answer: (B)

Explanation: This is very simple application of stars and bars. Since we want atleast k balls in each bag, so first we put kn balls into bags, k balls in each bag. Now we are left with m – kn balls, and we have to put them into n bags such that each bag may receive 0 or more balls. So applying theorem 2 of stars and bars with m – nk stars and n bars, we get number of ways to be m−kn+n-1 Cn−1. So option (B) is correct.


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