The constant term in the expansion of (2x+1x)^7(3x^2)^? *(five-term multinomial variant)* …

Question

The constant term in the expansion of $\left(2x+\dfrac1x\right)^7(3x^2)^{?}$ *(five-term multinomial variant)* is \_\_\_\_\_\_.

Answer: 1080

Accepted Answer

Using the multinomial general term $\dfrac{5!}{n_1!n_2!n_3!}(2x)^{n_1}(x^{-7})^{n_2}(3x^2)^{n_3}$ with $n_1+n_2+n_3=5$; for the constant term, $n_1+2n_3=7n_2$ and $n_1+n_2+n_3=5$. The only solution is $n_1=1,n_2=1,n_3=3$.

$$\text{Constant term}=\frac{5!}{1!1!3!}(2)^1(3)^3=20\times2\times27=1080$$

Solutions · 1