Question If 7 dice are rolled what is the probability that e

Question: If 7 dice are rolled, what is the probability that each of the six numbers will appear at least once?

Solution

There are 6767 sequences of 77 tosses, all equally likely.

We now count the sequences in which all the numbers appear. The number that appears twice can be chosen in 66 ways. For every such choice, where it appears can be chosen in (72)(72) ways. And the rest of the numbers can be put in the empty slots in 5!5! ways. So our probability is

[ 6 * [7C2] * 5! ] / 6^7

7C2 = 21

the probability is = 15120 / 279936 = 0.0540