Question

lf $(1+x{)}^n=\sum _{i=0}^n{C}_i{x}^i$ then the sum of the products of $C_{i}s$ taken two at a time is represented by $\sum _{0\le i<j\le n}{c}_i{c}_j$

• A. $2^n-\frac{\left(2n\right)!}{2(n!{)}^2}$
• B. $2^n+\frac{\left(2n\right)!}{2(n!{)}^2}$
• C. $\left(2^{{}^{2}n}\frac{\left(2n\right)!}{(n!{)}^2}\right)$
• D. $\frac{2^{2n}}{2(n!{)}^2}$

Answer 1

The correct option is C $\left(2^{{}^{2}n}\frac{\left(2n\right)!}{(n!{)}^2}\right)$

In some questions, substituting n=a positive number in both the question and the answer is the fastest way to achieve the correct option. Although there is always alternate options like writing the general term and splitting it to a format that can be solved but that takes long in a limited time paper. Try to put $n=2$ . In the end only option C remains.