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Q.
Find the sixth term of the expansion $(y^{1/2} + x^{1/3})^n$, if the binomial coefficient of the third term from the end is $45$.
Binomial Theorem
Solution:
Given expansion is $(y^{1/2} + x^{1/3})^n$
Since $T_6 = T_{5 + 1} =\,{}^nC_5(y^{1/2})^{n-5}(x^{1/3})^5\quad ...(i)$
Since binomial coefficient of third term from the end $=$
Binomial coefficient of third term from the beginning $= \,{}^nC_2$
$\therefore \,{}^{n}C_{2} = 45$
$\Rightarrow \frac{n\left(n-1\right)\left(n-2\right)!}{2!\left(n-2\right)!}= 45$
$\Rightarrow n \left( n - 1\right) = 90$
$\Rightarrow n^{2} - n - 90 = 0$
$\Rightarrow n = 10$
$\therefore $ From $\left(i\right), T_{6} = \,{}^{10}C_{5}\,y^{5/2} \,x^{5/3} $
$= 252\, y^{5/2} \cdot x^{5/3}$