4 players roll a die. They score 1 point if there is a matching pair. What are the probabilities of getting the possible scores.

For example: for the case where 2 people get 1, and the other two roll a 3 and a 4 respectively, then the teams score would be 1. Similarly, for the case that 3 people throw a 5 and the other one does not, the score would be 3, since there are 3 diﬀerent pairs of players that have the same number.

Proposed solution:

Imagine players as slots to fill with possibilities of rolling dice.

Score 0:

All 4 players roll different numbers. So 6*5*4*3 = 360 ways of getting score 0.

Score 1:

2 Players roll the same, other 2 are different from each other and the matching pair. So 6*1*5*4 = 120 ways of getting score 0.

Score 2:

2 pairs of players match. So 6*1*5*1 = 30 ways. The ones are the corresponding matches to the preceding rolls.

Score 3:

3 matches and 1 different so 6*1*1*5 = 30 ways again. The 4th person can have any one of the 5 choices left over.

Score 4:

All roll the same number so. 6*1*1*1 = 6 ways of getting score 4.

My problem is these all add up to 546, whereas I would imagine they'd have to add up to possibilities.

Any suggestions?