m identical balls are to be placed in n distinct bags. You are given that m >=…
2003
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.
\(\binom{m-k}{n-1}\)
- B.
\(\binom{m-kn+n-1}{n-1}\)
- C.
\(\binom{m-1}{n-k}\)
- D.
\(\binom{m-kn+n+k-2}{n-k}\)
Attempted by 3 students.
Show answer & explanation
Correct answer: B
Each bag must contain at least k balls. First place k balls into each of the n bags.
Balls used initially = kn.
Remaining balls = m - kn.
Now distribute these remaining m - kn identical balls among n distinct bags with no further restriction. If y_i is the number of extra balls in bag i, then
y_1 + y_2 + ... + y_n = m - kn, where each y_i >= 0.
By stars and bars, the number of non-negative integer solutions is
\(\binom{(m-kn)+n-1}{n-1}\).
Therefore, the number of ways is \(\binom{m-kn+n-1}{n-1}\).