Multinomial series based on binomial series

Question

I know that for

$(1+x)^k=1+kx+\dfrac{k(k-1)}{2!}x^2+\dfrac{k(k-1)(k-2)}{3!}x^3+\dfrac{k(k-1)(k-2)(k-3)}{4!}x^4...$

I wish to find the multinomial series so that I can produce more Taylor series for them, so what is the formula for let say:

Trinomial: $(1+x+y)^k$

Quadrinomial: $(1+x+y+z)^k$

Please write them explicitly like I do, if you wish to include the binomial notation, please write explicitly a few terms first.

Answer 1

This is the standard multinomial series $(\sum_{i=1}^mx_i)^n =\sum_{n_i \ge 0, \sum_{j=1}^m n_j = n}\binom{n}{n_1, n_2, ..., n_m}\prod_{i=1}^m x_i^{n_i} $ where $\binom{n}{n_1, n_2, ..., n_m} =\dfrac{n!}{\prod_{i=1}^m n_i!} $.

See, for example, https://en.wikipedia.org/wiki/Multinomial_theorem