Why is this equality with sums and monomials true?

Question

Consider $$(x_1 + ... + x_n)^k = \sum_{|\alpha| = k}c_{\alpha}x^{\alpha}$$ where $x^{\alpha} = x_{1}^{a_1}\cdots x_{n}^{a_n}$ and $|\alpha| = a_1 + ... + a_n$.

Why is this true? Is it something to do with the binomial sum?

Answer 1

You can consider it related to the binomial sum, as any multinomial, can be rewritten as nested binomials. However it's technical name is the multinomial sum. Consider the trinomial power $$(a+b+c)^3$$. It can be rewritten as $$(a+(b+c))^3$$ because of the associative property of addition, which is then a binomial power with $$b+c=d$$ Expanding once, we get:$$a^3+3da^2+3ad^2+d^3$$, which follows the binomial sum of exponents rule. Expanding the powers of d, will also not break the binomial exponent sum rule (d is a binomial after all). Therefore expanding it fully, the whole thing will not break the exponent sum rule. We then get:$$a^3+3ba^2+3ca^2+3ab^2+6abc+3ac^2+b^3+3cb^2+3bc^2+c^3$$