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If $x^{18}$ occurs in the rth term in the expansion of $\left(x^4+\frac{1}{x^3}\right)^{15}$, then what is the value of $r$ ?
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The correct answer is:
7
7
In the expansion of $\left(x^4+\frac{1}{x^3}\right)^{15}$, let $T_r$ is the $r^{\text {th }}$ term
$\begin{aligned}
T_r & ={ }^{15} C_{r-1}\left(x^4\right)^{15-r+1}\left(\frac{1}{x^3}\right)^{r-1} \\
& ={ }^{15} C_{r-1} x^{64-4 r-3 r+3}=15_{C_{r-1}} x^{67-7 r}
\end{aligned}$
$\mathrm{x}^{18}$ occurs in this term
$\Rightarrow 18=67-7 r \Rightarrow 7 r=49 \Rightarrow r=7 .$
$\begin{aligned}
T_r & ={ }^{15} C_{r-1}\left(x^4\right)^{15-r+1}\left(\frac{1}{x^3}\right)^{r-1} \\
& ={ }^{15} C_{r-1} x^{64-4 r-3 r+3}=15_{C_{r-1}} x^{67-7 r}
\end{aligned}$
$\mathrm{x}^{18}$ occurs in this term
$\Rightarrow 18=67-7 r \Rightarrow 7 r=49 \Rightarrow r=7 .$
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