# A six sided die is rolled six times. What is the probability that each side appears exactly once?

Sep 5, 2017

The probability is approximately 1.54%.

#### Explanation:

On the first roll, there are no restrictions. The die is allowed to be any of the 6 equally likely values. Thus the probability of not duplicating any numbers so far after roll 1 is $\frac{6}{6}$, or 100%.

For each subsequent roll, the number of "successful" rolls decreases by 1. For instance, if our first roll was a $\left[3\right]$, then the second roll needs to be anything but $\left[3\right]$, meaning there are 5 "successful" outcomes (out of the 6 possible) for roll 2. So, since each roll is independent of the previous rolls, we multiply their "success" probabilities together. The probability of rolling no repeats after two rolls is $\frac{6}{6} \times \frac{5}{6} = \frac{5}{6} ,$ which is about 83.3%.

Continuing this pattern, the third roll will have 4 "successful" outcomes out of 6, so we get

Pr("3 unique rolls") = 6/6 xx5/6 xx 4/6=55.6%

and then

Pr("4 unique rolls") = 6/6 xx5/6 xx 4/6 xx3/6=27.8%

Pr("5 unique rolls") = 6/6 xx5/6 xx 4/6 xx3/6xx2/6=9.26%

and finally

Pr("6 unique rolls") = (6 xx 5 xx 4 xx 3 xx 2 xx 1)/(6^6)=1.54%.