Pat will walk from intersection A to intersection B 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 A to B can Pat take that have the minimum possible length? Pick one:
A) 6
B) 8
C) 10
D) 14
E) 16
Consider this a as coordinate plane with A as origin and B as (2,3).   The shortest possible way is when you go take either right or up. (lefts and downs make repetitions and thus non shortest paths)   now, to get to (2,3) form (0,0) you need min 5 steps.   so you have to choose all the paths such that you have 2 steps out of 5 along x axis and getting to 3 of y axis.   The number of ways = 5C2 =10   (or alternately all paths that have 3 steps along y and takes you to 2 of x. .. 5C3 = 10)   Can you explicate your combinatorial method a bit more. For example, If I add another Avenue, eg, Avenue D, and another street, Street 5, I will have to go 3 steps right and 4 steps up to get to point C.   In order the length to be minimum Pat should only go UP and RIGHT: namely thrice UP and twice RIGHT.   So combination of UUURR: # of permutations of 5 letters out of which there are 3 identical U's and 2 identical R's is 5!/3!2!=10.   Answer: 10.
<result>C</result>
Pat will walk from intersection A to intersection B 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 A to B can Pat take that have the minimum possible length?