Question

Consider the expansion ${\left(x^2+\frac{1}{x}\right)}^{15}$.

Consider the following statements:
$1$. There are $15$ terms in the given expansion.
$2$. The coefficient of $x^{12}$ is equal to that of $x^3$.
Which of the statements is$/$are correct$?$

• A. $1$ only
• B. $2$ only
• C. Both $1$ and $2$
• D. Neither $1$ nor $2$

Answer 1

The correct option is B $2$ only

1. As the power is 15 so there would be 16

2. The genral term can be written as $T_{r+1}{=}^{15}{C}_r.x^{2(15-r)}.\frac{1}{x^r}{=}^{15}{C}_r.x^{30-3r}$

$T_{r+1}^{\prime }$ be the term for which power of x is 12

so $\Rightarrow$ $30-3r=12$ $\Rightarrow$ $r=6$

$T{"}_{r+1}$ be the term for which power of x is 3

so $\Rightarrow$ $30-3r=3$ $\Rightarrow$ $r=9$

The coefficent of $T_{r+1}^{\prime }$=${}^{15}{C}_6$ and coefficent of $T{"}_{r+1}$=${}^{15}{C}_9$ and as ${}^n{C}_x{=}^n{C}_{n-x}$

So there coefficent is same.