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. 

Solution

$$^nC_3 \, = \, ^nC_{12} \, \Rightarrow \, n \, = \, 15$$ 
$$\left ( x^2 \, + \, \dfrac{1}{x} \right )^{15}.$$  We have to find the term independent of x . It will be 11th term 
$$T_{11} \, = \, ^{15}C_{10} \, (x^2)^5 \, \left (\dfrac{1}{x}  \right )^{10} \, = \, ^{15}C_5 \, = \, 3003$$