Exercise · ZW-MULTINOMIAL-COEFFICIENT-009 · ★★★ Advanced
Nine distinct books are placed on 3 identical shelves, 3 books per shelf. How many distinct distributions are there?
Choose one option.
Show answer
- D. 280
Explanation
With labeled shelves there are \(\binom{9}{3,3,3}=1680\); since the shelves are identical, divide by \(3!=6\), giving 280.
Review the concept: Multinomial coefficient
Multinomial coefficient \(\binom{n}{k_1,\dots,k_m}\): the number of ways to split \(n\) distinct objects into \(m\) labeled groups of sizes \(k_1,\dots,k_m\), where \(k_1+\cdots+k_m=n\).
\[\binom{n}{k_1,\dots,k_m}=\frac{n!}{k_1!\,k_2!\cdots k_m!}\]
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