Suppose we have 15 different books, And we want these books to be split between 3 people.
How many ways are there such that each person will have exactly five books?
[Math] Suppose we have 15 different books, And we want these books to be split between 3 people.
combinationspermutations
Best Answer
There are $\binom{15}{5,5,5} = \frac{15!}{5!5!5!}$ ways to split 15 objects into 3 subsets of 5 elements each. (An alternative notation for multinomial coefficients is $(5,5,5)!$, as used by Wolfram)
So, the correct answer is $$\binom{15}{5}\binom{10}{5}\binom{5}{5}=\binom{15}{5,5,5}=\frac{1307674368000}{120^{3}}=756756$$
If you now multiply by the number of permutations of the three people, $3!$, you overcount, because you are distinguishing between the cases when, say, Alice gets books $b_1,b_2,b_3,b_4,b_5$ as the first person and the cases when she gets exactly the same books as the second person. As H.S correctly pointed out in a comment to another answer, if you split three books evenly among three people, you get $\binom{3}{1,1,1} = 6$ ways and not $\binom{3}{1,1,1}\cdot3!=36$ ways. What matters is who receives which books, not the order in which the books are distributed.
See also this similar question.