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📅 May 20, 2026 👤 scraface

May 20, 2026

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Solution: We want the number of onto functions from 7 birds to 3 bands...
\sum_{k=0}^{3} (-1)^k \binom{3}{k} (3 - k)^7
Compute terms:
$ k = 0 $: $ \binom{3}{0} \cdot 3^7 = 1 \cdot 2187 = 2187 $
$ k = 1 $: $ -\binom{3}{1} \cdot 2^7 = -3 \cdot 128 = -384 $
$ k = 2 $: $ \binom{3}{2} \cdot 1^7 = 3 \cdot 1 = 3 $
$ k = 3 $: $ -\binom{3}{3} \cdot 0^7 = -1 \cdot 0 = 0 $
Sum: $ 2187 - 384 = 1803 $, $ 1803 + 3 = 1806 $, $ 1806 + 0 = 1806 $
Thus, the number of valid frequency assignments is $ \boxed{1806} $.
\boxed{1806}

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