Question 44
Find (in terms of $n \geq 1$) the number of terms with odd coefficients after expanding the product:
$$
\prod_{1 \leq i\lt j \leq n}\left(x_{i}+x_{j}\right)
$$
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.
$$
\prod_{1 \leq i\lt j \leq n}\left(x_{i}+x_{j}\right)
$$
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.