# A six sided die is rolled six times. What is

A six sided die is rolled six times. What is the probability that each side appears exactly once?
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Step 1
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 [3], then the second roll needs to be anything but [3], 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 $\frac{6}{6}×\frac{5}{6}=\frac{5}{6},$, which is about $83.3%.$.
Step 2
Continuing this pattern, the third roll will have 4 "successful" outcomes out of 6, so we get
$Pr\left(\text{3 unique rolls}\right)=\frac{6}{6}×\frac{5}{6}×\frac{4}{6}=55.6%$
and then $Pr\left(\text{4 unique rolls}\right)=\frac{6}{6}×\frac{5}{6}×\frac{4}{6}×\frac{3}{6}=27.8%$
$Pr\left(\text{5 unique rolls}\right)=\frac{6}{6}×\frac{5}{6}×\frac{4}{6}×\frac{3}{6}×\frac{2}{6}=9.26%$
and finally $Pr\left(\text{6 unique rolls}\right)=\frac{6×5×4×3×2×1}{{6}^{6}}=1.54%.$.