Stanley's Saga

Probability Level 3

The protagonist of this story, Stanley, is standing peacefully at coordinates $(0, \space 0)$ without any concern in his life whatsoever.

Now, Stanley will move at coordinates $(3, \space 4)$ .

If Stanley only goes either up, or right, or up-right (corresponding vectors in their respective order: $\vec{u}(0, \space 1)$ , $\vec{r}(1, \space 0)$ , $\vec{ur}(1, \space 1)$ ), on how many ways can Stanley accomplish this tremendous stunt?