jmc

Counting the number of outcomes in which four 6-sided dice don't all show the same number would require some pretty delicate casework. However, counting all the outcomes in which four 6-sided dice do all show the same number is very easy: there are only 6 ways this can happen, namely all ones, all twos, all threes, all fours, all fives, and all sixes. So since there are $6^4$ total outcomes, we can conclude that $$P(\text{4 dice all show the same number})=\frac{6}{6^4}=\frac{1}{6^3}=\frac{1}{216}.$$Therefore, using the principle of complementary probabilities, we can conclude that $$P(\text{4 dice don’t all show the same number})=1-\frac{1}{216}=\boxed{\frac{215}{216}}.$$