Coefficient in the expansion of (2x-3y)^12

Text

Find the coefficient of the term $x^{7}y^{5}$ in the expansion of $(2x - 3y)^{12}$.

Solution

By the binomial theorem, the general term of the expansion of $(A+B)^{n}$ is $\binom{n}{k}A^{k}B^{n-k}$. Here $A=2x$, $B=-3y$, $n=12$; we want $x^{7}y^{5}$, so $k=7$ (and $n-k=5$): $\binom{12}{7}(2x)^{7}(-3y)^{5} = \binom{12}{7}\,2^{7}(-3)^{5}\,x^{7}y^{5}.$ We compute the numerical coefficient with $\binom{12}{7}=792$, $2^{7}=128$, $(-3)^{5}=-243$: $c = 792 \cdot 128 \cdot (-243) = \boxed{-24\,634\,368}.$