Place Cards in the Wind

A banquet hall has $31{,}415$ assigned seats, each with a guest's place card. A gust scatters all the cards, and a rushed usher lays them back down in a uniformly random arrangement, one card per seat. Let $A$ count the cards that landed back on their own seat and $B$ count the cards that did not. What is $\mathrm{Var}(B - A)$?