m identical balls are to be placed in n distinct bags. You are given that m = n and each bag must contain at least 1 ba...

Question

m identical balls are to be placed in n distinct bags. You are given that m >= n and each bag must contain at least 1 ball. In how many ways can the balls be distributed?

A. C(m-1, n-1)
B. C(m-n+1, n-1)
C. C(m-1, m-n)
D. C(m+n-1, m-n)

Accepted Answer

C(m-1, n-1) We have m identical balls and n labeled bags. Each bag must get at least 1 ball. Because balls are identical, only the count per bag matters, not which specific balls go where. This is a classic stars-and-bars distribution with a lower bound of 1 per bin.

m identical balls are to be placed in n distinct bags. You are given that m >= n and each bag must contain at least 1 ball. In how many ways can the balls be distributed? A. C(m-1, n-1) B. C(m-n+1, n-1) C. C(m-1, m-n) D. C(m+n-1, m-n)