This is a great candidate for applying the Mississippi Rule
The rule is useful for arranging a group of items in which some of the items are identical.
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!)]

-----NOW ONTO THE QUESTION--------
To get from point a to point b, we must travel UP (U) two times, and travel RIGHT (R) 4 times
In other words, we want to determine the number of DIFFERENT ways to arrange 2 U's and 4 R's
let's apply the above rule.

There are 6 letters in total
There are 2 identical U's
There are 4 identical R's
Total number of possible arrangements = 6!/[(2!)(4!)]
= 15

Answer:D