Permutation & Combinations - Distribution

Question

The number of ways in which n distinct things can be distributed among n people so that at least one person does not get anything is 232. Find n. I think every object has (n-1) option. So (n-1)^n=232. But this gives the answer as 4.4(not a whole number) whereas the answer is 4.

Where am I going wrong?

I assume the $n$ people are distinct, is the same true for the things? — Asinomás, Aug 15 '15 at 15:53
Your formula is finding the distributions so that Bob does not get anything. You are missing all the distributions in which Bob gets something but some other person does not. — Asinomás, Aug 15 '15 at 15:57

score 4 · Accepted Answer · answered Aug 15 '15 at 15:55

4

There are $n!$ ways in which everyone gets a thing. There are $n^n$ distributions in total.

Hence the formula for what you want is $n^n-n!$.

We start trying with $n=4$ and get $4^4-4!=232$. So we have found it.

answered Aug 15 '15 at 15:55

Asinomás

105,651

Nicely explained. – Shashank Gupta Aug 15 '15 at 15:59
Simple explanation. +1 – Shailesh Aug 15 '15 at 16:06

Permutation & Combinations - Distribution

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