Example:
There are $3$ containers, with capacity $4, 2,$ and $3$ items, respectively. How many way to arrange $6$ items on those containers?
Answer:
container size $= 4, 2, 3$
number of items to be arranged $= 6$
possible arrangement with container size of:
$4, 2, 3 \leftarrow$ container size
$1, 2, 3 \leftarrow$ 1st arrangement
$2, 1, 3 \leftarrow$ 2nd arrangement
$2, 2, 2 \leftarrow$ 3rd arrangement
$3, 0, 3 \leftarrow$ 4th arrangement
$3, 1, 2 \leftarrow$ 5th arrangement
$3, 2, 1 \leftarrow$ 6th arrangement
$4, 0, 2 \leftarrow$ 7th arrangement
$4, 1, 1 \leftarrow$ 8th arrangement
$4, 2, 0 \leftarrow$ 9th arrangement
as such there are $9$ possible arrangements. So the answer is $9$.
How to calculate that, instead of doing it manually?