Question

The co-efficient of y in the expansion of $(y^2+c/y{)}^5$ is?

• A. $10c^3$
• B. $20c^2$
• C. $10c$
• D. $20c$

Answer 1

The correct option is A $10c^3$

$(y^2+c/y{)}^5{=}^5{C}_0{\left(\frac{c}{y}\right)}^0{\left(y^2\right)}^{5-0}{+}^5{C}_1{\left(\frac{c}{y}\right)}^1{\left({}^{y^2}\right)}^{5-1}+...{+}^5{C}_5{\left(\frac{c}{y}\right)}^5{\left(y^2\right)}^{5-5}$

$=\sum _{r=0}^5$ ${}^5{C}_r$ ${\left(\frac{c}{y}\right)}^r{\left(y^2\right)}^{5-r}$

We need cofficinet of y $\Rightarrow 2(5-r)-r=1$

$\Rightarrow 10-3r=1$

$\Rightarrow r=3$

So, cofficient of $y{=}^5{C}_3.C^3$

$=10C^3$