Correctly calculating permutations and combinations without duplicate patterns Given 16 balls each numbered 1 through 16, and 5 glass tubes numbered 1 through 5; how many ways are there to slot all 16 balls into the glass tubes, selected one at a time, with the only condition that each slot should always have at least 1 ball? The ball and glass numbers matter.

We assume that order of balls in the tubes matters. 
Line up the $16$ balls in some order.  There are $16!$ ways to do this. There are $15$ interball gaps. We choose $4$ of them to place a separator into in the usual Stars and Bars style. This can be done in $\binom{15}{4}$ ways.
Thus the total number of ways is $16!\binom{15}{4}$. The idea generalizes.
Remark: Numerically, this gives the same result as the recursion of dREaM.
