I think your solution is wrong.Since there are two 5's and three 4's we have to divide your solution by 2!*3! and also we have to multiply your answer by 5! because we can arrange the 5 different groups in 5! ways.

This is exactly the solution given in my textbook. But I don't agree with it because multiplying the solution by 5! makes no sense. It would mean that students who need 5 books are provided with 4 books and vice-versa.

@Lokesh Sharma
–
You are forgetting that the students are distinct. There is a difference between:

How many ways are there to split 4 books into a group of 3 and a group of 1?

and

How many ways are there to distribute 4 books to 2 students, such that one student receives 3 books and the other student receives 1 book?

In one case, the order of the groups matter, and in the other case, the order doesn't. This is where multiplying by $5!$ comes in, because order matters.

@Calvin Lin
–
u mean to say that the answer is
22! / {(5!⋅5!⋅4!⋅4!⋅4!)* 5!} due to 5 students ,,,,,,,,,
and if in the case of 5 groups or 5 sets it will be 22!/(5!⋅5!⋅4!⋅4!⋅4!)

@Pravas Patra
–
No, it would be $\frac {22!}{5! 5! 4! 4! 4! 2! 3!}$ in case of dividing into groups and $\frac {22!}{5! 5! 4! 4! 4! 2! 3!} * 5!$ in above asked student case.

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## Comments

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TopNewest${22 \choose 5,5,4,4,4} = \frac{22!}{5! \cdot 5! \cdot 4! \cdot 4! \cdot 4!} = 5 646 383 542 800$

It's a multinomial coefficient, i.e. the way of divide 22 different objects into 5 different groups of 5, 5, 4, 4 and 4.

Wikipedia: Multinomial Coefficient

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I think your solution is wrong.Since there are two 5's and three 4's we have to divide your solution by 2!*3! and also we have to multiply your answer by 5! because we can arrange the 5 different groups in 5! ways.

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This is exactly the solution given in my textbook. But I don't agree with it because multiplying the solution by 5! makes no sense. It would mean that students who need 5 books are provided with 4 books and vice-versa.

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and

In one case, the order of the groups matter, and in the other case, the order doesn't. This is where multiplying by $5!$ comes in, because order matters.

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$\frac {22!}{5! 5! 4! 4! 4! 2! 3!}$ in case of dividing into groups and $\frac {22!}{5! 5! 4! 4! 4! 2! 3!} * 5!$ in above asked student case.

No, it would beLog in to reply

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I think you are correct..

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Thanks... Got it. I was getting the same answer but was not sure.

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