Question

The number of $4\times 4$ skew symmetric matrices that can be formed using exactly four $0$'s, six $3$'s and six $-3$'s is

Answer 1

For skew symmetric matrix $a_{ij}=-a_{ji}$ and diagonal elements $=0$
Let $A$ be the matrix then
$A=\left[\begin{array}{cccc}0& a& b& c\\ -a& 0& d& e\\ -b& -d& 0& f\\ -c& -e& -f& 0\end{array}\right]$
Each element above diagonal can be filled in $2$ ways either $3$ or $-3$.
The number such matrix possible
$=2^6=64$