Question

If $C_0,C_1,C_2,\cdots ,C_n$ denote the binomial coefficients in the expansion of ${(1+x)}^n$ , then the value of the expression $C_0{C}_r+C_1{C}_{r-1}+C_2{C}_{r-2}+\cdots +C_{n-r}{C}_r+$ equals

• A. $\frac{2n!}{(2n-r)!(2n+r)!}$
• B. $\frac{n!}{(n-r)!(n+r)!}$
• C. $\frac{2n!}{(n-2r)!(n+2r)!}$
• D. none of these

Answer 1

The correct option is D none of these

Consider the following series.
${}^m{C}_0{}^n{C}_r+{}^m{C}_1{}^n{C}_{r-1}+...{}^m{C}_r{}^n{C}_0$
=Coefficient of $x^r$ in $(1+x{)}^m.(x+1{)}^n$
=coefficient of $x^r$ in $(1+x{)}^{m+n}$
$={}^{m+n}{C}_r$
In the above case, $r=n-r$ and $m=n$
Hence
${}^{2n}{C}_{n-r}$
$=\frac{2n!}{(2n-(n-r\left)\right)!(n-r)!}$
$=\frac{2n!}{(n+r)!(n-r)!}$