You work as a driver for a ride$-$sharing company. The city you drive in has all east$-$west roads named alphabetically and all north$-$south roads named numerically. All blocks in the city are perfectly aligned in a grid.

$($a$)$ Your first passenger needs to be picked up on the corner of Maryland and $3$rd$,$ then dropped off at the

corner of Sanders and $7$th$.$ How many shortest paths along the grid are there to drive your passenger to

their destination?

$($b$)$ You have another passenger to pick up on the corner of Parkway and $5$th$.$ How many paths in the previous

part pass through this intersection?