How to picture the number of $p$-permutations of $t$ things with $k$ kinds, where $n_1, n_2, n_3, \cdots , n_k$ = the number of each kind of thing?

Question

Some students can't even grok this problem statement, as they are (informationally) overloaded by the number of variables : $p, t, k, n_1, \cdots, n_k \;$. Kindly improve my picture, or draw a better picture?

I match variables to the first letter of the mathematical object. E.g. $N$umber of each $\color{fuchsia}{k}$ind $= n_\color{fuchsia}{k}$.

Answer 1

Imagine you are a typesetter manually placing sorts – metal blocks containing raised mirrored characters on one face – into a line. You have $n_i$ sorts of character $i$, and $k$ different characters available, so $t=\sum_{i=1}^kn_i$. Then you want to count the number of $p$-character strings you can make with your limited resources.