How many ways are there of rolling 6 different coloured 6-sided dice so that exactly 3 different values are showing? Any help with this problem would be greatly appreciated. I have tried solving this problem like this:
1st die: 6 choices,
2nd die: 5 choices,
3rd die: 4 choices,
4th die: 3 choices,
5th die: 3 choices,
6th die: 3 choices
But this isn't getting me anywhere! Thanks.

Do this problem in three steps:


*

*Choose 3 of the 6 values

*Partition the six dice into 3 non-empty groups

*Assign the 3 values from step 1 to the 3 groups from step 2
The total number of ways to complete these steps is ${6\choose 3}\times {6\brace 3}\times 3!=20\times 90\times 6=10800.$
