Question

Consider the expansion ${\left(x^2+\frac{1}{x}\right)}^{15}$. What is the sum of the coefficient of the middle terms in the given expansion?

• A. $C(15,9)$
• B. $C(16,9)$
• C. $C(16,8)$
• D. None of the above

Answer 1

The correct option is D None of the above

Given expansion is ${(x^2+\frac{1}{x})}^{15}$,

The total number of terms are $16$,

$\therefore$ there are two middle terms in the expansion which are,

$\left(\begin{array}{c}15\\ 7\end{array}\right),\left(\begin{array}{c}15\\ 8\end{array}\right)$,

$\therefore$ the sum of the coefficients is $\left(\begin{array}{c}15\\ 7\end{array}\right)+\left(\begin{array}{c}15\\ 8\end{array}\right)=\left(\begin{array}{c}16\\ 7\end{array}\right)$

The formula used is $\left(\begin{array}{c}n\\ r\end{array}\right)+\left(\begin{array}{c}n\\ r-1\end{array}\right)=\left(\begin{array}{c}n+1\\ r-1\end{array}\right)$