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

12john
Thread moved from the technical forums to the schoolwork forums
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