Given positive integers $$r > 1, n > 2$$   and that the co - efficient of $${\left( {3r} \right)^{th}}\,{\text{and }}{\left( {r + 2} \right)^{th}}$$    terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$  are equal. Then

Solution :
Given that $$r$$ and $$n$$ are +ve integers such that $$r > 1, n > 2$$
Also in the expansion of $${\left( {1 + x} \right)^{2n}}$$
co - eff. of $${\left( {3r} \right)^{th}}$$  term = co - eff. of $${\left( {r + 2} \right)^{th}}$$  term
$$\eqalign{ & \Rightarrow \,{\,^{2n}}{C_{3r - 1}} = {\,^{2n}}{C_{r + 1}} \cr & \Rightarrow \,\,3r - 1 = r + 1\,\,{\text{or }}3r - 1 + r + 1 = 2n \cr & \Rightarrow \,\,r = 1\,\,{\text{or }}2r = n \cr & {\text{But }}\,r > 1 \cr & \therefore \,\,n = 2r \cr} $$