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If the coefficients of $(2 r+1)^{\text {th }}$ term and $(r+1)^{\text {th }}$ term in the expansion of $(1+x)^{42}$ are equal then $r$ can be
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Verified Answer
The correct answer is:
14
We have, $(1+x)^{42}$
$$
\begin{gathered}
T_{2 r+1}={ }^{42} C_{2 r} x^{2 r} \\
T_{r+1}={ }^{42} C_r x^r
\end{gathered}
$$
Coefficient of $(2 r+1)^{\text {th }}$ and $(r+1)^{\text {th }}$ term are equal
$$
\begin{aligned}
& \therefore \quad{ }^{42} C_{2 r}={ }^{42} C_r \\
& \therefore \quad 2 r+r=42 \quad\left[\because{ }^n C_x={ }^n C_y \Rightarrow x+y=n\right] \\
& \Rightarrow \quad r=14 \\
&
\end{aligned}
$$
$$
\begin{gathered}
T_{2 r+1}={ }^{42} C_{2 r} x^{2 r} \\
T_{r+1}={ }^{42} C_r x^r
\end{gathered}
$$
Coefficient of $(2 r+1)^{\text {th }}$ and $(r+1)^{\text {th }}$ term are equal
$$
\begin{aligned}
& \therefore \quad{ }^{42} C_{2 r}={ }^{42} C_r \\
& \therefore \quad 2 r+r=42 \quad\left[\because{ }^n C_x={ }^n C_y \Rightarrow x+y=n\right] \\
& \Rightarrow \quad r=14 \\
&
\end{aligned}
$$
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