Question

Coefficient of $x^n$ in $({}^n{C}_0+{}_1^{C}x+{}^n{C}_2{x}^2......{}^n{C}_n{x}^n)\times ({}^n{C}_0+{}^n{C}_1\times .....{}^n{C}_n{x}^n)$ is

• A.
• B.
• C.
• D.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A

B

C

D

$({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2.....{}^n{C}_n{x}^n)$ is the expansion of $(1+x{)}^n$ so the given term is $(1+x{)}^n\times (1+x{)}^n=(1+x{)}^{2n}$
Coefficient of $x^n$ in the expansion of $(1+x{)}^{2n}is{}^{2n}{C}_n\Rightarrow$ (B)

We get the coefficient of $x^n$ when we multiply, ${}^n{C}_0$ from 1st sum and ${}^n{C}_n$ from 2nd sum.

We get the same by multiplying ${}^n{C}_{1}and{}^n{C}_{n-1},{}^n{C}_{2}and{}^n{C}_{n-2}......{}^n{C}_{n}and{}^n{C}_0$. We are considering the terms with $x^n$ when we multiply each term.



$\Rightarrow$ coefficient of $x^n={}^n{C}_0{}^n{C}_n+{}_1^C{}^n{C}_{n-1}.......{}^n{C}_n{}^n{C}_0$
$\Rightarrow$ C
We can write ${}^n{C}_{r}as{}^n{C}_{n-r}$
$\Rightarrow$ coefficient of $x^n$ = ${}^n{C}_0\times {}^n{C}_0+{}^n{C}_1\times {}^n{C}_1.....{}^n{C}_n\times {}^n{C}_n$
$=({}^n{C}_0{)}^2+({}^n{C}_1{)}^2+.......({}^n{C}_n{)}^2$
$\Rightarrow$(D)
$={\sum }_{r=0}^n({}^n{C}_r{)}^2$ ---(A)