Find the coefficients of $x^{n} y^{n}$ in the expansion of
[(1 + x) (1 + y) (x + y)] $^{n}$

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Solution

$(1 + x)^{n} = C_{0} + C 1 x + C_{2} x^{2} + . . . . + C_{n} x^{n}$
$(y + 1)^{n} = C_{0} y^{n} + C_{1} Y^{n - 1} C_{2} Y^{n - 2} + . . + C_{n}$
Note ( y + 1)
$(x + y)^{n} = C_{0} x^{n} + C_{1} x^{n - 1} y + C_{2} x^{n - 2} y^{2} + C_{n} y^{n}$
If will be multiply them , each vertical column will be the term of $x^{n} y^{n}$
$∴ (1 + x)^{n} (1 + y)^{n} (x + y)^{n}$ the coefficient of $x^{n} y^{n}$ is
$C_{0}^{3} + C_{1}^{3} + C_{2}^{3} + . . . + C_{n}^{3}$

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