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?

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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.

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