jmc

There are $6^6$ different possible rolls exhibited by the six dice. If the six dice yield distinct numbers, then there are $6$ possible values that can appear on the first die, $5$ that can appear on the second die, and so forth. Thus, there are $6!$ ways to attain $6$ distinct numbers on the die. The desired probability is $\frac{6!}{6^6}=\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{6\cdot 6\cdot 6\cdot 6\cdot 6\cdot 6}=\frac{20}{6\cdot 6\cdot 6\cdot 6}=\frac{5}{2^2\cdot 3^4}=\boxed{\frac{5}{324}}$.