What is the formula for permutations of multiple kinds in combinatorics?

[12john]

My combinatorics professor has a MA, PhD from Princeton University. On our test, she asked

What's the explicit formula for 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 ?

I handwrote, but transcribed in Latex, my answer below.

To deduce the formula for all the unique permutations of length $l$ of $\{n_1,n_2,...,n_k\}$, we must find all combinations $C=\{c_1,c_2,...,c_k\}$ where $0 \leq c_k \leq n_k$, such that
$\sum_{i=1}^k c_i=l$.

What we need, is actually the product of the factorials of the elements of that combination:
${\prod_{i=1}^k c_i!}$

Presuppose that the number of combinations is J. Then to answer your question, the number of permutations is
$$= \sum_{j=1}^J \frac{l!}{\prod_{i=1}^k c_i!}
= \sum_{c_1+c_2+...+c_k=l} \binom{l}{c_1,c_2, \cdots ,c_n},$$
as a closed form expression with a Multinomial Coefficient. *QED.*

How can I improve this? What else should I've written? Professor awarded me merely 50%. She wrote

Your answer is correct, but your solution is too snippy. You need to elaborate.

 

[Mark44]

I handwrote, but transcribed in Latex, my answer below.

I edited your LaTeX. At this site we use MathJax, which has to be delimited by either pairs of # characters (for inline TeX) or pairs of $ characters (standalone).
I also removed all the color stuff. We prefer that you use a minimum of extra color, bolding, italics, etc.

How can I improve this? What else should I've written?

What she wrote was "You need to elaborate." The best explanation would come from your instructor.