Expansion of cube of four terms

Question

How do I get to know the pattern that appears when we open the cube of four terms ?

For example, how do I get to know this pattern?

$(a+b+c+d)^3=\sum a^3 $ +6 $\sum abc$ + 3$\sum b a^2$

Answer 1

In $(a+b+c+d)^3$ we know that all terms will exactly have a degree $3$. Now all distinct types of terms possible are $\sum_{cyc}a^3, \sum_{cyc}a^2b, \sum_{cyc}abc$. Coefficients can be evaluated using counting principles.

For example, coefficient of $a^3$ will be $\binom{3}{3}$, that of $a^2b$ will be $\binom{3}{2}\binom{1}{1}$, and that of $abc$ will be $\binom{3}{1} \binom{2}{1} \binom{1}{1}$

To conclude, you can write:

$$(a+b+c+d)^3 = \binom{3}{3}\sum_{cyc}a^3+ \binom{3}{2} \binom{1}{1}\sum_{cyc}a^2b+ \binom{3}{1} \binom{2}{1} \binom{1}{1}\sum_{cyc}abc$$