Problem related to multinomial expansion.

Question

How do I expand $(a_1+a_2+a_3+.....+a_k)^3$ where $a_i \in \Bbb R $ for $i = 1,2,...,k $?

Answer 1

Here you go:

$$(a_1+a_2+...+a_k)^3=(a_1+a_2+...+a_k)\cdot(a_1+a_2+...+a_k)^2=(a_1+a_2+...+a_k)\cdot\big(\sum_{i=1}^{k}a_i^2+2\sum_{1\leq i<j\leq k}a_ia_j\big)=\sum_{i=1}^{k}a_i^3+3\sum_{1\leq i<j\leq k}a_ia_j(a_i+a_j)+6\sum_{1\leq i<j<l\leq k}a_ia_ja_l$$