1. We throw 12 identical balls (indistinguishable) randomly into 9 bins. What is

Question

1. We throw 12 identical balls (indistinguishable) randomly into 9 bins. What is the probability that no bin is empty? Assume the bins are distinguishable (i.e. numbered 1 through 9).

2. Let n1,n2,n3,...,nt be positive integers. Show that if n1 + n2 + ... + nt t + 1 balls are placed into t bins, then for some i {1,2,...,t} , the i-th bin contains at least ni balls.

For Question 1 I know that there are "20 choose 12" possibilities to get 12 indistinguishable balls into 9 distinguishable bins. But I don't know the probability that no bin is empty.

For Question 2, I'm completely lost.

Explanation / Answer

There are (summation i=1 to k [n_i] ) - k + 1 balls. We can place (summation i=1 to k [n_i-1])=(summation i=1 to k [n_i]) - k Pigeonhole principle. QED

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