Let $S^n$ be the $n$-sphere. Then the unordered configuration space $B(S^m,2)=F(M,2)/\Sigma_2$ is the total space of a line bundle over $\mathbb{R}P^m$, i.e. we have a fibre bundle

$$ \mathbb{R}\to B(S^m,2)\to \mathbb{R}P^m. $$

How to compute the total Stiefel-Whitney class of the tangent bundle of $B(S^m,2)$ $$ w(TB(S^m,2))? $$