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The coefficient of x^n y^n in the expansion of [(1+x)(1+y)(x+y)]^n is

Option: 1

\sum_{r=0}^n C_r


Option: 2

\sum_{r=0}^n C_r^2


Option: 3

\sum_{r=0}^n C_r^3


Option: 4

none of these


Answers (1)

best_answer

\begin{aligned} & {[(1+x)(1+y)(x+y)]^n=(1+x)^n(1+y)^n(x+y)^n } \\ & \therefore \quad(1+x)^n=C_0+C_1 x+C_2 x^2+\ldots . .+C_n x^n \ldots . (1)\\ &(1+y)^n \equiv(y+1)^n=C_0 y^n+C_1 y^{n-1}+C_2 y^{n-2}+\ldots .+C_n \ldots . (2)\\ &(x+y)^n=C_0 x^n+C_1 x^{n-1} y+C_2 x^{n-2} y^2+\ldots . .+C_n y^n\ldots . (3) \end{aligned}

If we multiply equations (1), (2), and (3) side by side, on R.H.S. product of terms in each column will contain x^n y^n.
\therefore \quad$ Coefficient of $x^n y^n$ in $(1+x)^n(y+1)^n(x+y)^n$ is $$ \begin{aligned} C_0^3 & +C_1^3+C_2^3+\ldots . .+C_n^3 \\ & =\sum_{r=0}^n C_r^3 \end{aligned}

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