Pat will walk from intersection X to intersection Y
along a route that is confined to the square grid of
four streets and three avenues shown in the map
above. How many routes from X to Y can Pat take
that have the minimum possible length ?
ans: 10

Pat will walk Up 3 times and Right 2 times. So, the path is similar to a 5 letter word with 3Us and 2Rs.
Number of distinct ways = 5!/(3!*2!) = 10.