If we define Pat's path in a block-by-block manner, we can see that any route from X to Y will consist of 3 UPS and 2 RIGHTS.
So for example, if we let U represent walking one block UP, and let R represent walking one block RIGHT, one possible path is URURU.
Another possible path is UUURR
Another possible path is UURUR

So our question becomes, "In how many different ways can we arrange 3 U's and 2 R's?"

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When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:

If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....]

So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:
There are 11 letters in total
There are 4 identical I's
There are 4 identical S's
There are 2 identical P's
So, the total number of possible arrangements = 11!/[(4!)(4!)(2!)]
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Now let's apply the MISSISSIPPI rule to arranging 3 U's and 2 R's
There are 5 letters in total
There are 3 identical U's
There are 2 identical R's
So, the total number of possible arrangements = 5!/[(3!)(2!)] = 10

Answer: C

Cheers,
Brent