Question

Consider the expansion of (3x + 2y)^4 What is the coefficient of the xy^3 term?

Answer 1

The given expression is:

$(3x+2y)^4$

This can be expanded as:

$\begin{gathered} (3x+2y)^4=4C0(3x)^4(2y)^0+4C1(3x)^3(2y)^1 \\ +4C2\mleft(3x\mright)^2\mleft(2y\mright)^2+4C3\mleft(3x\mright?^{})^1(2y)^3+4C4(3x)^0(2y)^4 \end{gathered}$

Note that:

$\begin{gathered} \text{nCr}=\text{ }\frac{n!}{(n-r)!r!} \\ 4C0=\text{ 1} \\ 4C1=4 \\ 4C2=6 \\ 4C3=4 \\ 4C4=1 \end{gathered}$

The expansion then becomes:

$(3x+2y)^4=\text{ }81x^4+216x^3y+216x^2y^2+96xy^3+16y^4$