The multinomial theorem

Terms. The multi-indices $(i,j,k)$ with sum $3$: $(3,0,0)$, $(0,3,0)$, $(0,0,3)$, $(2,1,0)$, $(2,0,1)$, $(1,2,0)$, $(0,2,1)$, $(1,0,2)$, $(0,1,2)$, $(1,1,1)$.

Coefficients. $\frac{3!}{i!j!k!}$: $1$ for $(3,0,0)$, $3$ for permutations of $(2,1,0)$, $6$ for $(1,1,1)$.

Result. $(a+b+c)^3=a^3+b^3+c^3+3a^2b+3a^2c+3ab^2+3b^2c+3ac^2+3bc^2+6abc$.

Conclusion. The multinomial expansion.