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In the expansion of $\left(1+3 x+3 x^2+x^3\right)^{2 n}$, the term which has greatest binomial coefficient, is
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$(3 n+1)$ th term
$\because$ Middle term has greatest binomial coefficient.
In the expansion of $\left(1+3 x+3 x^2+x^3\right)^{2 n}$
$=\left((1+x)^3\right)^{2 n}=(1+x)^{6 n}$
$\because 6 n$ is even
So, middle term of $(1+x)^{6 n}=T\left(\frac{6 n}{2}+1\right)$ $=T_{(3 n+1)}=(3 n+1)$ th term.
In the expansion of $\left(1+3 x+3 x^2+x^3\right)^{2 n}$
$=\left((1+x)^3\right)^{2 n}=(1+x)^{6 n}$
$\because 6 n$ is even
So, middle term of $(1+x)^{6 n}=T\left(\frac{6 n}{2}+1\right)$ $=T_{(3 n+1)}=(3 n+1)$ th term.
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