Question

The coefficient of $x^n{y}^n$ in the expansion of $\left\{\right(1+x\left)\right(1+y\left)\right(x+y){\}}^n$ is

• A. $\sum _{r=0}^n{C_r}^2$
• B. $\sum _{r=0}^n{C_r}^3$
• C. $\sum _{r+s=0}^n{}^n{C}_r\cdot {{}^n{C}_s}^2$
• D. None of these

Answer 1

The correct option is B $\sum _{r=0}^n{C_r}^3$

We have,
$\left\{\right(1+x\left)\right(1+y\left)\right(x+y){\}}^n$
$=(C_0+C_{1}x+C_2{x}^2+\dots +C_n{x}^n)\times (C_0{y}^n+C_1{y}^{n-1}+C_2{y}^{n-2}+\dots +C_{n-1}y+C_n)\times (C_0{x}^n+C_1{x}^{n-1}y+\dots +C_{n-1}x{y}^{n-1}+C_n{y}^n)$

If we multiply them , each vertical column will be the term of $x^n{y}^n$
$\therefore$ Coefficient of $x^n{y}^n$ on R.H.S
$={C_0}^3+{C_1}^3+{C_2}^3+\dots +{C_n}^3$
$=\sum _{r=0}^n{C_r}^3$