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Home » So, \( t = 2 \) or \( t = 4 \). - Get link 4share

So, \( t = 2 \) or \( t = 4 \). - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 This is a classic inclusion-exclusion problem. The total number of assignments (without restriction) is $5^8$, since each of the 8 species has 5 choices. But we must subtract the assignments where at least one device is not used.

📌 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 $

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