When six dice are rolled, the number of different numbers which can appear can range from 1 to 6. In how many different ways can you have EXACTLY 4 different numbers?
When 6 dice are rolled each of them can turn up a number from 1 to 6. The number of ways in which exactly 4 different numbers turn up has to be determined.
If exactly 4 different numbers are to turn up, 4 of the dice have distinct numbers. The other two can either turn up with the same number as one of the 4 dice or each can turn up with a different number but one which is common to 2 of the other 4 dice.
The number of ways in which this can happen is now described. Four of the dice turn up a different number. This is possible in 6*5*4*3 = 360 ways. If the other two dice have the same number there are four possible combinations. If the two dice have different numbers there are 4*3 = 12 possibilities. Adding up gives the total number of possibilities as 360*(4 + 12) = 360*16 = 5760
When 6 dice are rolled exactly four different numbers can turn up in 5760 ways.
It seems like something is missing here,
first we are able to choose any 4 numbers out of 6 so we have 6C4. next we choose 4 dice out of 6 so 6C4. those 4 dice can be one of the 4 selected numbers in 4! ways. For the last two dice we can have the 4 numbers in 4*4 ways. so the total seems like 6C4*6C4*4!*4*4=86400 (correct me if you see an error in my thinking)