How can I improve my answer deducted 50%, even though it's correct?

[StackExchange]

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

What's the formula for the number of [math]p[/math] permutations of [math]n_1 + n_2 + \cdots + n_{k - 1} + n_k[/math] things with [math]k[/math] kinds, where [math]n_1, \cdots , n_k[/math] = the number of each kind of thing ?

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

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

What we need, is actually the product of the factorials of the elements of that combination:
[math]\color{limegreen}{\prod_{i=1}^k c_i!}[/math]

Presuppose that the number of combinations is J. Then to answer your question, the number of permutations is
[math]= \sum_{j=1}^J \frac{l!}{\color{limegreen}{\prod_{i=1}^k c_i!}} = \sum_{c_1+c_2+...+c_k=l} \binom{l}{c_1,c_2, \cdots ,c_n},[/math]
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 merely wrote

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

 

[R.M.]

I'm in my 22nd year of teaching and recently completed my masters degree, but given all that experience and knowledge, I have no idea what is inside ANY teacher's head when he/she grades any work. Not a single one of my students would ever go to my colleague and ask why I wrote a -1 next to an answer. Go directly to the source, or to the T.A. (if there is one, this person probably graded it in the first place).

 

[Dr.Peterson]

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

I concur with the opinion that you have to ask the grader, not outsiders. At the very least, "snippy" is an odd word to have used (I think of it as reflecting an attitude more than mere brevity!), so I don't even know for sure what it means here.

But in addition, it seems to me that (a) you used a variable not mentioned in the problem, namely l, instead of p; (b) you claim to be "deducing the explicit formula" and a "closed form expression", which is not what I would call your summation; (c) you used an index, j, in your summation but not in the summand. Apparently those aren't considered to be reasons to call the answer wrong. But certainly more explanation would be good; have you tried writing it more fully?

On the other hand, the question as you quote it just asks for a formula, not a derivation. So I'm not sure what is really expected.

I also see that you got the same answer here.