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Home » #### \(\frac10713, \frac6213\) - Simpleprint

#### \(\frac10713, \frac6213\) - Simpleprint

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Okay, let's tackle thisQuestion: An ornithologist is tracking 8 species of birds using GPS tags, and each bird can be tagged with one of 5 different tracking devices. If each species must use at least one device, how many ways can the ornithologist assign devices to species?

📌 Solution: We are assigning 5 distinct tracking devices to 8 bird species such that each species gets at least one device. This is equivalent to counting the number of onto functions from a set of 8 species to a set of 5 devices, with the condition that no device is left unused.

📌 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

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