examPAL DS Forum Expert David - Ask Me Anything about GMAT DS

Hello indu1954,
Apologies for the delay in response, if you want me to address a question quickly please tag me in it.

This is a difficult question. The most straightforward (easy-to-understand) solution is to split into cases.

All 5 cards have the same color:
Pick all the blue cards (5C5 = 1) and arrange them (5!)
Pick all the red cards (5C5 = 1) and arrange them (5!)
Total 5! + 5! = 240 ways

Ratio is 1:4
Pick 1 blue card (5C1 = 5) and 'arrange it' (1! = 1). Pick 4 red cards (5C4 = 5) and arrange them (4!). Choose if the order is blue --> red or red --> blue (2 ways).
Do the same for 1 red, 4 blue.
Total 5*5*4!*2 + 5*1!*5*4!*2 = 100*4! = 2400 ways.

Ratio is 2:3
Pick 2 blue cards (5C2 = 10) and arrange them (2! = 2). Pick 3 red cards (5C3 = 10) and arrange them (3!). Choose if the order is blue --> red or red --> blue (2 ways).
Do the same for 2 red, 3 blue.
Total 10*2*10*6*2 times 2 = 200*4! = 4800 ways.

So there are 4800 + 2400 + 240 = 7440 ways in total
The total number of ways to pick 5 cards out of 10 and arrange them is 10C5 * 5! = 10*9*8*7*6

Then the answer is 7440/10*9*8*7*6 = 744/9*8*7*6 = (800 - 56)/9*8*7*6 = (100 - 7)/9*7*6 = 93/9*7*6 = 31/3*7*6= 31/126.
Answer (C).