Probability - can you explain me the answer

Aquarium fish are 5 blue, 7 yellow and three red fish. you take Out at random 3 fish. What is the probability to have one of each color?

I was told that the answer is -

(5 * 7 * 3) / 15ncr3

can you explain it to me?

Ran -

I always suggest you look at the Numerator and Denominator separately ...

NUMERATOR: There are 5 ways to pick a blue fish and 7 ways to pick a yellow fish and 3 ways to pick a red fish. In total, there are 5 x 7 x 3 = 105 ways to pick a blue, yellow and red fish.

DENOMINATOR: The sample space picking 3 fish out of 15 is: 15 ways to pick the first fish, 14 ways to pick the 2nd fish and 13 ways to pick the 3rd fish. In total there are 15x14x13 = 2730 ways to pick 3 fish "if order matters". But since order does not matter, divide by 3! = 6. So, the denominator = 2730/6 = 455

P = 105/455 = 3/13

Hope that helped

First, let us calculate the number of ways one can take three fishes from 5 + 7 + 3 = 15. Since order doesn't matter in this case (e.g, picking a blue, red, and yellow fish in that order is the same as picking, say, a red, blue, and yellow fish in that order), there are C(15, 3) ways to do this.

Now, suppose that the draw order is blue, yellow, and red. Since the order doesn't matter, we can assume this without a lack of generality. There are 5 ways to pick a blue fish on the first draw (since there are 5 blue fish), 7 ways to pick a yellow fish on the second draw, and 3 ways to pick a red fish on the third draw. Thus, by the counting principle, there are 5*7*3 ways to pick 3 random fish in such a way that each fish is a different color.

Therefore, the probability is (5*7*3)/C(15, 3), as required.

I hope this helps!