That result is for six indistinguishable rooks (I am trusting you on that, I have not checked it). So the final answer is this times the number of ways that the six can be coloured such that two are red and four are blue, which is 15 ways.

(Given any acceptable configuration of indistinguishable rooks, they can be numbered in order from the top left in the order they appear when read from left to right down the rows. Then we have the problem of assigning the colours to the numbered rooks for this configuration )

CB