I want to distribute $15$ distinguishable balls into $8$ baskets such that the first $5$ ones contain $10$ balls. Are my calculations correct?
Let's say I have 10 balls, there are $ 5^{10} $ to distribute them amongst the first 5 ones. I have $5$ more balls left and $3$ more cells, and so I can distribute those $5$ in $ 3^5 $ ways. Therefore, the number is:
$ \binom{15}{10}\cdot5^{10}\cdot3^5 $