probability Permutations and Combinations

Question

How many ways are there to put 30 identical balls in to 5 different boxes if each box must contain at least 3 balls?

Answer 1

Hint:

Put three balls in each box before continuing the process. You now have $15$ balls remaining that you wish to distribute into your $5$ different boxes, possibly not putting any additional balls in a box beyond the three initially placed there.

Alternatively, put two balls in each box before continuing the process. You now have $20$ balls remaining that you wish to distribute to your $5$ different boxes but require putting at least one additional ball in each box.

This is a standard problem type which can be solved using the technique of stars-and-bars.

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The gist of stars and bars is that we relate an arrangement of the balls into the boxes to a sequence of $*$'s and $|$'s, for example $**|***|*|**$ describing four boxes, the first of which getting two balls, the second getting three, the third getting one, and the fourth getting two. Counting the number of arrangements of an appropriate number of $*$'s and an appropriate number of $|$'s for our scenario is then equivalent to counting the number of outcomes to our original scenario. More information can be found in the link above.