A 15 sided dice is rolled 1000 times. Let k1,k2,k3,k4,..k15 denote the number of times 1,2,3...15 appears.
How can I compute the following expected value :$$E( (k_1 k_2 k_3 k_4)^5).$$
My attempts::
I tried using multinomial theorem , then using the moment-generating function and then the stirling numbers but failed miserably.
Edit: Note that I was able to find $E( (k_1 k_2 k_3 k_4))$, a general formula in terms of falling factorial but not any further .