Problem Poker dice is played by simultaneously rolling 5 dice. How many ways can we form "1 pair", "2 pairs"?
For one pair, I got the answer right away. First I consider there are 5 spots for 5 dice. Then I pick 2 places out of 5, which means there are 3 places left, so we have to choose 3 out of 3 which is 1 way. Hence, I have: $${{5}\choose{2}} \cdot 6 {{3}\choose{3}} \cdot 5 \cdot 4 \cdot 3 = 3600.$$ However, I couldn't figure out why I got two pairs wrong. First, I pick 2 places for the first pair, then its rank. Next, 2 places for the second pair, and its rank. Since there is only 1 place left, I pick the rank for the last dice. $${{5}\choose{2}} \cdot 6 {{3}\choose{2}} \cdot 5 \cdot 4 \cdot 3 = 3600.$$ But the correct answer is 1800, which means I need to divide by a factor of 2. I guess that might be the order of two pairs can be switched, but I wonder is there a better way to count it? I'm so confused! Any idea?