Can someone help me with real life application of the Multinomial Coefficient? I have been studying the Multinomial Coefficient theorem for some time now. But I don't get how one can apply it in real life aside being used for making of selections and the number of times rearrangements of items can be made. I am very interested in the topic and I'll be happy if i can get more real life applications.
Thanks
 A: A very simple example is you use the multinomial when you want to find the number of ways to arrange the letters in a word like "MISSISSIPPI", in which you have one "M", four "I"s, four "S"s, and two "P"s. The number of distinct ways to arrange the letters is $\frac{11!}{1!4!4!2!}=\binom {11}{4,2,2}$. You also use it when you want to find $(x_1+x_2+...x_n)^k$.
A: Typical real-life examples might arise in probability and statistics.  For example, you might find this case study interesting.  While the details involving Bayesian analysis might be beyond you at this stage, the executive summary is: they analyze presidential election polls in 2008, using models that involve a multinomial distribution.  That is, if you take a random sample of $n$ voters
from a population with given probabilities of responses (say $p_1$ for candidate A, $p_2$ for candidate B, $p_3$ for candidate C, $p_4$ undecided), the probability of getting $n_1$ responses of A, $n_2$ of B, $n_3$ of C, $n_4$ undecided is $$ {n \choose n_1, n_2, n_3, n_4} p_1^{n_1} p_2^{n_2} p_3^{n_3} p_4^{n_4}$$
The statistician wants to estimate these probabilities, and how they changed over time.
