Hi guys, I'm really bad at probabilities, so here goes my problem:

We simulate a poker game with 5 identical dice (with 6 different faces each). What are the probabilities of rolling the dice and getting :

1) 5 different numbers

2) A pair

3) Two pairs

4) Three of a kind

I got #1 and #2 right; I'm however having problems with the latter two, so I'm not too sure if I have the right way of thinking the first two.

For the first one, I did the following reasoning: There are 6 ways to get a first card, 5 ways to get a second card, 4 ways to get a third card, 3 ways to get a fourth card and 2 ways to get a fifth card. Thus the probability is

.

For the second one : there are 6 ways to choose which card will be doubled, thus 1 way to get the second (doubled) card, and then 5 ways to get the third card, 4 to get the fourth and 3 to get the fifth. The probability is therefore

.

However, I have two contradicting reasonings for #3, none of which are correct according to my answer book. I can think like this: there are 6 ways to choose which one will be doubled first, then there are 5 ways to choose which one will then be doubled, and 4 ways to choose the last card, bringing the probabilities to

.

I also have this reasoning: there are

ways to get the two cards who will be doubled, and 4 ways to choose the last card, giving probabilities of

.

I'm thoroughly confused. Can anybody help?

Thanks!