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12john

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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

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

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

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.

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