I'm a little bit confused about what the meaning of multinomial distributions, at least from what i've gleaned from the wikipedia page on Multinomial Distribution. In essence, a multinomial distribution is the generalized form of a binomial distribution. That is, outcomes are independent, however there are k possible outcomes, each with k success, which gives the probability mass function:
$Mult(n, p_1, p_2, p_3, ..., p_n) = {n \choose x_1, x_2, ... x_n} p_1^{x_1} ...p_n^{x_n} $
However, shouldn't evaluating only $X_1$ give a binomial distribution since the multinomial is just a generalization of it? But throwing that in, we get: $Mult(n, p_1) = {n \choose x_1} p_1^{x_1}$ which can't possibly turn into $Bin(n, p) = {n \choose x} p^x(1-p)^{n-x}$
Am I missing something in my understanding of a multinomial distribution? And if so, how do we get the binomial from the multinomial equation then?