The coefficient of $x^5$ in the expansion of $\left(x^2+2 x+3\right)^5$, is

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Aicharya · Accepted Answer

Given, $\left(3+2 x+x^2\right)^5$ $=\Sigma \frac{n !}{(p !)(q !)(r !)}(3)^p(2 x)^q\left(x^2\right)^r$...(i) Where, $\quad p+q+r=n=5$ For $x^5$, we have, $q+2 r=5$ $\begin{aligned} & \therefore \quad r=0, q=5, p=0 \ & r=1, q=3, p=1 \ & r=2, q=1, p=2 \end{aligned}$ Putting the above values in Eq. (i), we have $\begin{aligned} & \frac{5 !}{0 ! 5 ! 0 !}(3)^0(2)^5(1)^0+\frac{5 !}{1 ! 3 ! 1 !}(3)^1\left(2^3(1)^1\right. \ & \quad=\frac{5 !}{2 ! 1 ! 2 !}(3)^2(2)^1(1)^2 \ & \quad=32+20 \times 3 \times 8+30 \times 9 \times 2 \ & \quad 32+480+540=1052 \end{aligned}$

The coefficient of \(x^5\) in the expansion of \(\left(x^2+2 x+3\right)^5\), is

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