Why does $(a + b)^3$ expand to $a^3 + 3ab^2 + 3a^2b + b^3$ 
Why does $(a + b)^3$ expand to $a^3 + 3ab^2 + 3a^2b + b^3$

Why does this work? I am confused as to why this happens. 

In overly pedantic detail:
\begin{align}
(a+b)^3
 &= (a+b)[(a+b)(a+b)] \tag{by definition} \\
 &= (a+b)[a(a+b) + b(a+b)] \tag{distribution} \\
 &= (a+b)[a^2 + ab + ba + b^2] \tag{distribution} \\
 &= (a+b)[a^2 + 2ab + b^2] \tag{commutativity} \\
 &= a[a^2 + 2ab + b^2] + b[a^2 + 2ab + b^2] \tag{distribution} \\
 &= a^3 + a2ab + ab^2 + ba^2 + b2ab + b^3 \tag{distribution} \\
 &= a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3 \tag{commutativity} \\
 &= a^3 + 3a^2b + 3ab^2 + b^3. \tag{collect like terms}
\end{align}
