### Binomial Theorem

goals
If the coefficient of 4th and 13th terms in the expansion of $\left(x^2 \, + \, \dfrac{1}{x}\right)^n$ be equal then find the term which is independent of x.
Find out the sum  of the coefficient  in the  expansion  of the binomial $(5p \, - \, 4q) ^n$  where  n is  a + ive  integer .
Express $\dfrac{14}{-5}$ as a rational number with numerator $56$.
Multiply the binomials.
$\left( {2x + 5} \right)$ and $\left( {4x - 3} \right)$

If $\displaystyle C_{r}=^{n}C_{r}$ then to evaluate the expansion $\displaystyle A=\sum \sum_{0\leq r\leq s\leq n}^{}$ $\displaystyle C_{r}C_{s}$, We make use of $\displaystyle \sum_{r=0}^{n}C_{r}^{2}=^{2n}C_{n}$ and the expansion of $\displaystyle \left ( \sum_{r=0}^{n}C_{r} \right )^{2}$. On the basis of above information answer the following questions.