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Home » You Won’t BELIEVE What Skylanders Unlocked—Guess What’s Coming Next! - Dyverse

You Won’t BELIEVE What Skylanders Unlocked—Guess What’s Coming Next! - Dyverse

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Question:** A youth mentorship program has 10 students and 6 mentors. In how many ways can each student be assigned to exactly one mentor, if each mentor must be assigned at least one student?

📌 This is a classic surjective mapping problem: assigning 10 distinguishable students to 6 distinguishable mentors such that no mentor is left unassigned.

📌 This is the number of **onto** functions from a 10-element set to a 6-element set, which is:

📌 \sum_{k=0}^{6} (-1)^k \binom{6}{k} (6 - k)^{10}

📌 This uses the inclusion-exclusion principle:

📌 $ k = 0 $: $ \binom{6}{0} \cdot 6^{10} = 1 \cdot 60466176 = 60466176 $

📌 $ k = 1 $: $ -\binom{6}{1} \cdot 5^{10} = -6 \cdot 9765625 = -58593750 $

📌 $ k = 2 $: $ \binom{6}{2} \cdot 4^{10} = 15 \cdot 1048576 = 15728640 $

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