Ten distinct balls are distributed into four distinct boxes labelled…

Question

Ten distinct balls are distributed into four distinct boxes labelled 1, 2, 3, 4 so that no box is empty. In how many of these distributions does box 1 contain an even number of balls (where 0 is not possible since no box is empty, so this means 2, 4, 6, or 8)?

Accepted Answer

We count onto distributions (surjections) of 10 distinct balls to 4 distinct boxes with box 1 getting an even, positive count.

Let box 1 receive exactly k balls (k in {2,4,6,8}). Choose which balls: C(10,k). The remaining 10-k balls must go to boxes 2,3,4 with none of those three empty: that is the number of surjections from a (10-k)-set onto 3 labelled boxes, S = 3! * Stirling2(10-k, 3) = 3^(10-k) - 3*2^(10-k) + 3.

k=2: C(10,2)=45, remaining 8 onto 3 surjective = 3^8-3*2^8+3 = 6561-768+3 = 5796 -> 45*5796 = 260820.
k=4: C(10,4)=210, remaining 6 onto 3 = 729-192+3 = 540 -> 210*540 = 113400.
k=6: C(10,6)=210, remaining 4 onto 3 = 81-48+3 = 36 -> 210*36 = 7560.
k=8: C(10,8)=45, remaining 2 onto 3 = 8-24+3 = ... = 6-... compute: 3^2-3*2^2+3 = 9-12+3 = 0 -> 45*0 = 0.

Sum = 260820 + 113400 + 7560 + 0 = 381780.

Final answer: 381780

Ten distinct balls are distributed into four distinct boxes labelled…

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