19. [10] Find (in terms of $n \geq 1$ ) the number of terms with odd coefficients after expanding the product:
$$
\prod_{1 \leq i<j \leq n}\left(x_{i}+x_{j}\right)
$$
e.g., for $n=3$ the expanded product is given by $x_{1}^{2} x_{2}+x_{1}^{2} x_{3}+x_{2}^{2} x_{3}+x_{2}^{2} x_{1}+x_{3}^{2} x_{1}+x_{3}^{2} x_{2}+2 x_{1} x_{2} x_{3}$ and so the answer would be 6 .
Answer: $n!$
Note that if we take $(\bmod 2)$, we get that
$$
\prod_{1 \leq i<j \leq n}\left(x_{i}+x_{j}\right) \equiv \prod_{1 \leq i<j \leq n}\left(x_{j}-x_{i}\right)=\operatorname{det}(M)
$$
where $M$ is the matrix with $M_{i j}=x_{i}^{j-1}$. This is called a Vandermonde determinant. Expanding this determinant using the formula
$$
\operatorname{det}(M)=\sum_{\sigma} \prod_{i=1}^{n} x_{\sigma(i)}^{i-1},
$$
where the sum if over all $n$ ! permutations $\sigma$, gives the result.