$$r$$ and $$n$$ are positive integers $$r > 1, n > 2$$   and co - efficient of $${\left( {r + 2} \right)^{th}}$$  term and $$3{r^{th}}$$ term in the expansion of $${\left( {1 + x} \right)^{2n}}$$  are equal, then $$n$$ equals

Solution :
$$\eqalign{ & {t_{r + 2}} = {\,^{2n}}{C_{r + 1}}{x^{r + 1}};{t_{3r}} = {\,^{2n}}{C_{3r - 1}}{x^{3r - 1}} \cr & {\text{Given}}{{\text{ }}^{2n}}{C_{r + 1}} = {\,^{2n}}{C_{3r - 1}}; \cr & \Rightarrow \,{\,^{2n}}{C_{2n - \left( {r + 1} \right)}} = {\,^{2n}}{C_{3r - 1}} \cr & \Rightarrow \,\,2n - r - 1 = 3r - 1 \cr & \Rightarrow \,\,2n = 4r \cr & \Rightarrow \,\,n = 2r \cr} $$