We know that,

Total number of relation both symmetric and reflexive $=2^{\frac{n^2-n}{2}}$, where $n$ is number of elements in given set,

Also, total number of symmetric relation $=2^{\left(\frac{{\mathrm{n}}^2+\mathrm{n}}{2}\right)}$

Then number of symmetric relation which are not reflexive will be,

$=2^{\frac{n(n+1)}{2}}-2^{\frac{n(n-1)}{2}}$

$=2^{10}-2^6$

$=1024-64$

$=960$