How many ways are there to distribute $15$ distinguishable objects into $5$ distinguishable boxes so that the boxes have one, two, three, four, and five objects in them ?
I tried to solve it as =>
$C(15,1) * C(14,2) * C(12,3) * C(9,4) * C(5,5)$ and
since placing total number of objects in each box matters here ,i.e, objects being placed in boxes in the count $1,2,3,4,5$ is different from $3,1,5,4,2$ and so on.
So, i multiplied it by $5!$
Hence, final answer I think should be
$5! * C(15,1) * C(14,2) * C(12,3) * C(9,4) * C(5,5)$
I don't have answer for this question. So, is my approach right ?
Best Answer
Yes. That looks alright.
You counted the ways to partition the objects into groups of distinct sizes $1,2,3,4,5$, and then ways to arrange those groups into the boxes. That is what you wanted to count, and how you could count it.
Also written as $5!\dbinom{15}{5,4,3,2,1}$ using the multinomial coefficient notation.