Question

Find the coefficient of $x^{n-2}$ in
$({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2.....{}^n{C}_n{x}^n)\times ({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2.....{}^n{C}_n{x}^n)$

• A. ${}^n{C}_0\times {}^n{C}_{n-2}+{}^n{C}_1\times {}^n{C}_{n-3}.....{}^n{C}_n{}^n{C}_2$
• B. ${}^n{C}_0\times {}^n{C}_2+{}^n{C}_1\times {}^n{C}_3.....{}^n{C}_{n-2}\times {}^n{C}_n$
• C. ${\sum }_{r=0}^{n-2}{}^n{C}_r\times {}^n{C}_{n-2-r}$
• D. ${\sum }_{r=0}^n{}^n{C}_r\times {}^n{C}_{r-1}$

Answer 1

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

${\sum }_{r=0}^{n-2}{}^n{C}_r\times {}^n{C}_{n-2-r}$

In the expansion of
$({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2.....{}^n{C}_n{x}^n)\times ({}^n{C}_0+{}^n{C}_{1}x.....{}^n{C}_n{x}^n)$
We get the terms with $x^{n-2}$ when we multiply
$\left({}^n{C}_{0}and{}^n{C}_{n-2}{x}^{n-2}\right),\left({}^n{C}_{1}xand{}^n{C}_{n-3}{x}^{n-3}\right).....\left({}^n{C}_{n-2}{x}^{n-2}and{}^n{C}_0\right)$
From first sum and 2nd sum
⇒ The coefficient will be sum of each coefficient.
= ${}^n{C}_0\times {}^n{C}_{n-2}+{}^n{C}_1\times {}^n{C}_{n-3}.....{}^n{C}_{n-2}\times {}^n{C}_0$
This is what option (A) is. If we modify the terms using the property ${}^n{C}_r={}^n{C}_n-r$
We get, ${}^n{C}_0{}^n{C}_2+{}^n{C}_1\times {}^n{C}_3+.....{}^n{C}_{n-2}\times {}^n{C}_0$
This is option (B)
Option (C) is the summation written with variable. When we expand (C) we get (B) and (A).