4A. Calculate the coefficient of $x^{n} y^{n}$ if the expression

$$
(1+x)^{n}(1+y)^{n}(x+y)^{n}
$$

is expanded according to the Binomial formula.
Solution. According to the binomial formula

$$
(1+x)^{n}=\sum_{k=0}^{n}\binom{n}{k} x^{k},(1+y)^{n}=\binom{n}{k} y^{n-k} \quad \text { and }(x+y)^{n}=\sum_{k=0}^{n}\binom{n}{k} x^{n-k} y^{k} .
$$

The terms in the product $(1+x)^{n}(1+y)^{n}(x+y)^{n}$ where $x^{n} y^{n}$ appears are

$$
\binom{n}{k} x^{k}\binom{n}{k} x^{n-k} y^{k}\binom{n}{k} y^{n-k}, k=0,1,2, \ldots, n
$$

Therefore, the desired coefficient is $\binom{n}{0}^{3}+\binom{n}{1}^{3}+\ldots+\binom{n}{n}^{3}$.