Question

The number of different n×n symmetric matrices with each element being either 0 or 1 where n =5.

Answer 1

You have $5$ free entries along the diagonal. Since $a_{ij}=a_{ji}$, you only need to worry about how many choices are available for the triangular part either above or below the diagonal. These parts both contain $4+3+2+1=10$ entries.

So the number of possibly symmetric binary matrices is $2^5\times2^{10}=2^{15}$.