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# If the second ,third and fourth term in the expansion of $(x+a)^n$ and $240,720$ and $1080$ respectively,then the value of $n$ is

$\begin{array}{1 1}(A)\;15\\(B)\;20\\(C)\;10\\(D)\;5\end{array}$

Toolbox:
• $T_{r+1}=nC_r a^{n-r} b^r$
We have
$nC_1x^{n-1}a^1=240$------(1)
$nC_2x^{n-2}a^2=720$------(2)
$nC_3x^{n-3}a^3=1080$------(3)
Eliminating $x$ and $a$ we get
$\large\frac{(nC_1x^{n-1}a^1)(nC_3x^{n-3}a^3)}{(nC_2 x^{n-2} a^2)^2}=\frac{240\times 1080}{(720)^2}$
$\Rightarrow \large\frac{nC_1nC_3}{(nC_2)^2}=\frac{1}{2}$
$\Rightarrow n.\large\frac{n(n-1)(n-2)}{6}.\frac{2}{n(n-1)}.\frac{2}{n(n-1)}=\frac{1}{2}$
$\Rightarrow \large\frac{2(n-2)}{3(n-1)}=\frac{1}{2}$
$\Rightarrow 4n-8=3n-3$
$\Rightarrow n=5$
Hence (D) is the correct answer.