Problem with probabilities

[NukeG]

(see pic for context)

So if the Question was simpler, the probability of say a specific number it would be 1-5. But this question is asking for the probability of pulling 3 specific numbers (as only 3 of the five will give the desired outcome).

So first pick, 3 out of 5 chance of hitting one of the possible numbers needed.. and then If successful, 2 out of 4 on the second pick. Finally 1 out of 3 last pick.
So 3/5 x 2/4 x 1/3 - in which I get 10%. Which isn't even a possible answer. I know I'm misunderstanding this, can someone explain to me how to work it out?

 

[Dr.Peterson]

(see pic for context)

So if the Question was simpler, the probability of say a specific number it would be 1-5. But this question is asking for the probability of pulling 3 specific numbers (as only 3 of the five will give the desired outcome).

So first pick, 3 out of 5 chance of hitting one of the possible numbers needed.. and then If successful, 2 out of 4 on the second pick. Finally 1 out of 3 last pick.
So 3/5 x 2/4 x 1/3 - in which I get 10%. Which isn't even a possible answer. I know I'm misunderstanding this, can someone explain to me how to work it out?

What are "the possible numbers needed"? And doesn't the probability of "success" on the second depend on which one you picked first?

I'd approach this by first listing all the possible successful sets of three, and then using combinations.