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steam family sharing not same household - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Let $ A_i $ be the set of assignments where device $i$ is not used, for $ i = 1 $ to $ 5 $.

📌 Then the number of onto functions is:

📌 \sum_{k=0}^{5} (-1)^k \binom{5}{k} (5 - k)^8

📌 $ k = 0 $: $ (-1)^0 \binom{5}{0} \cdot 5^8 = 1 \cdot 1 \cdot 390625 = 390625 $

📌 $ k = 1 $: $ (-1)^1 \binom{5}{1} \cdot 4^8 = -1 \cdot 5 \cdot 65536 = -327680 $

📌 $ k = 2 $: $ (-1)^2 \binom{5}{2} \cdot 3^8 = 1 \cdot 10 \cdot 6561 = 65610 $

📌 $ k = 3 $: $ (-1)^3 \binom{5}{3} \cdot 2^8 = -1 \cdot 10 \cdot 256 = -2560 $

📌 $ k = 4 $: $ (-1)^4 \binom{5}{4} \cdot 1^8 = 1 \cdot 5 \cdot 1 = 5 $

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