Question

Find the sixth term of expansion of $(y^{\frac{1}{2}}+x^{\frac{1}{3}}{)}^n$, If third term from end is $45$.

• A. $252x^{\frac{5}{2}}{y}^{\frac{5}{3}}$
• B. $252y^{\frac{1}{2}}{x}^{\frac{1}{3}}$
• C. $252y^{\frac{5}{2}}{x}^{\frac{5}{3}}$
• D. $14y^{\frac{5}{2}}{x}^{\frac{5}{3}}$

Answer 1

The correct option is C $252y^{\frac{5}{2}}{x}^{\frac{5}{3}}$

Here, the general term, $T_{r+1}{=}^n{C}_r{y}^{\frac{r}{2}}{x}^{\frac{n-r}{3}}$

Total terms $=n+1$

$3^{rd}$ term from last is $(n+1)-3+1$ or $(n-1)$ term from first.

Now, $T_{(n-2)+1}{=}^n{C}_{n-2}{y}^{\frac{n-2}{2}}{x}^{\frac{2}{3}}{T}_{n-1}{=}^n{C}_2{y}^{\frac{n-2}{2}}{x}^{\frac{2}{3}}$

The coefficient of $(n-1)$ term is 45.

So, ${}^n{C}_2=45\Rightarrow \frac{n(n-1)}{2}=45\Rightarrow n=10$

Now, the $6^{th}$ term is

$T_6{=}^{10}{C}_5{y}^{\frac{10-5}{2}}{x}^{\frac{5}{3}}$

$=252.y^{\frac{5}{2}}{x}^{\frac{5}{3}}$