No child left without a toy

Question

Six different toys are given out to three children. Each toy is handed independently to one of the three children, chosen uniformly at random (so a child could receive anywhere from 0 to 6 toys).

What is the probability that every child receives at least one toy?

Accepted Answer

There are 3^6 = 729 equally likely ways to assign the toys. We need the number of assignments that leave no child empty-handed, a surjection from 6 toys onto 3 children. Count the "bad" assignments (someone empty) with inclusion–exclusion. Let Ai be the event child i gets nothing. - Each single Ai = 2^6 = 64; there are 3 of them. - Each pair Ai \cap Aj = 1^6 = 1 (all toys to the one remaining child); there are 3 pairs. - All three empty is impossible. Assignments with at least one empty child: 3\cdot 64 - 3\cdot 1 = 192 - 3 = 189. So the surjections number 729 - 189 = 540, and P(	ext{everyon…

No child left without a toy

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