[Math] 6 shirts, 4 pants, 7 scarves, choosing randomly and then probability of picking red shirt and blue scarf

Question

If I have 6 different color shirts (red, green, blue, yellow, white, black), 4 different color pants (blue, brown, black, grey) and 7 different color scarves (red, green, blue, yellow, white, black, purple) and then I randomly choose 4 shirts, 2 pants and 3 scarves (no repetition, order doesn't matter) from this set. Then, what is the probability that I would get a red shirt and blue scarf? I think the total amount of combinations would be $\dbinom{6}{4} * \dbinom{4}{2} * \dbinom{7}{3} $ but don't know where to go from there.

Answer 1

To count the favorable cases, observe that if you select a red shirt and a blue scarf among the four selected shirts, two selected pants, and three selected scarves, you must select three of the other five available shirts, two of the four available pants, and two of the other six available scarves. Therefore, the number of favorable cases is

$$\binom{5}{3}\binom{4}{2}\binom{6}{2}$$

Hence, the probability of selecting a red shirt and a blue scarf when four shirts, two pants, and three scarves are chosen from six different color shirts, four different color pants, and seven different color scarves is

$$\frac{\dbinom{5}{3}\dbinom{4}{2}\dbinom{6}{2}}{\dbinom{6}{4}\dbinom{4}{2}\dbinom{7}{3}}$$