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Q. In the expansion of $\left(3^{- x / 4}+3^{5 x / 4}\right)^{ n }$ the sum of binomial coefficient is $64$ and term with the greatest binomial coefficient exceeds the third by $( n -1)$, the value of $x$ must be

Binomial Theorem

Solution:

To get sum of coefficients put $x=0$. Given that sum of coefficients is
$2^{ n }=64$
or $n =6$
The greatest binomial coefficient is ${ }^{6} C _{3}$.
Now given that
$T _{4}- T _{3}=6-1=5$
$\Rightarrow { }^{6} C _{3}\left(3^{- x / 4}\right)^{3}\left(3^{5 x / 4}\right)^{3}-{ }^{6} C _{2}\left(3^{- x / 4}\right)^{2}\left(3^{5 x / 4}\right)^{4}=5$
which is satisfied by $x=0$.