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Home » Let’s compute how many ways to place 3 non-adjacent G’s in 5 positions. - Carbonext

Let’s compute how many ways to place 3 non-adjacent G’s in 5 positions. - Carbonext

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 A conservationist is deploying GPS trackers on 7 endangered birds, assigning each to one of 3 operational frequency bands. How many assignments ensure that each band is used by at least one bird?

📌 Solution: We want the number of onto functions from 7 birds to 3 bands. Using inclusion-exclusion:

📌 \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 $

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