Point sequences reaching four all in a tennis game

In a tennis game the score starts at $0$–$0$ and each point adds one to the winner's tally. A player who reaches $4$ points first wins the game, unless the score has been level at $3$–$3$, after which a player must lead by $2$ to win. In how many different point-by-point sequences can the score reach $4$–$4$?

$40$. To reach $4$–$4$ the game must pass through $3$–$3$ (otherwise someone would have won at $4$–$x$ with $x\le2$), and there are $\binom63=20$ ways to get there. From $3$–$3$ the next two points must be split, in either order: $20\cdot2=40$.