It's the same as the number of distinguishable arrangement of the row of 6
letters BBRRWW. 

That's because For any distinguishable arrangement of those 6 letters, we 
can always put the left shoe of each color left of the right one of the same
color.  

Since there are 6 letters, the 2 B's are indistinguishable, the 2 R's are
indistinguishable and the 2 W's are indistinguishable.   

So the answer is  

Edwin

The total number of permutations of 6 items is  6! = 720.


In this set of all permutations,  S(6), exactly HALF of all permutations has the "leftBLUE" item on the left from "rightBLUE" item.


So, regarding blue shoes, we have the subset in  S(6)  of   permutations, 
where red shoes are in the right order.


Let denote this subset as  S(6,B).



In the subset  S(6,B),  exactly HALF of its permutations has the "leftRED" item on the left from "rightRED" item.


So, regarding red AND blue shoes, we have the subset in  S(6,R)  of   = 180 permutations, 
where BOTH  red AND blue shoes are in the right order.


Let denote this subset as  S(6,B,R).



In the subset  S(6,R,B),   exactly HALF of its permutations has the "leftWHITE" item on the left from "rightWHITE" item.


So, regarding red, blue and white shoes, we have the subset in  S(6,R,B)  of   = 90 permutations, 
where all  red AND blue AND white shoes are in the right order.



It gives the final answer.