Number of ways to distribute n different toys among n children so that any one child gets no toy ?
I tried using "star and bar method" and deduced that i need to calculate the number of arrangements possible for the string "111…20" where there are (n-1) different 1's,one 0(since 1 child gets no toy),one 2(because 1 child gets 2 toys).
However i'm not being able to reach the final answer.Please help.
[Math] Number of ways to distribute n different toys among n children so that any one child gets no toy
combinatorics
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Best Answer
Both the toys and the children are distinguishable, so this is not a stars and bars problem: it would be a stars and bars problem if the toys were indistinguishable.
HINT: There are $n$ ways to choose the child who is to receive no toy. There are $\binom{n}2$ ways to form a pair of toys to be given to the lucky child, and there are $n-1$ ways to choose who is to get that pair. Now there are $n-2$ single toys to be distributed to the remaining $n-2$ children so that each gets one toy. Can you put the pieces together now to come up with the final result?