UR grid

I will solve using "The complementary method",

The number of all ways to go from $(0,0)$ to $(3,3)$ is $\binom{6}{3}=20$ ,

We have to subtract the ways that passes throught the two red dots , using " Multiplication principle " :

The number of ways from $(0,0)$ to $(1,1)$ multipied by the number of ways from $(1,1)$ to $(2,2)$ multipied by the number of the ways from $(2,2)$ to $(3,3)$ , the result is $\binom{2}{1}\binom{2}{1}\binom{2}{1}= 8$

So we get that the answer is $20-8=12$

Alternatively, there are four ways to pass through only the first red dot, four ways to pass through only the second red dot, and four ways to pass through no red dots.

$4+4+4=12$