<s>[INST] Probability Of Rolling A Strictly Increasing Sequence On A Six-Sided Die 
By rolling a six-sided die 6 times, a strictly increasing sequence of
  numbers was obtained, what is the probability of such an event?

I have no ideas on how to attack this. It says, an increasing sequence for the $6$ times were obtained. So this means, that on the first roll, you get $1$, and on second you get $2$, on the third you get $3$, and so on. The total sample space is given by $6^6$ and the probability of obtaining the set $\{1,2,3,4,5,6\}$ is $\frac{1}{6^6}$? Please help.
 [/INST] For each roll of the die, there are $6$ possible outcomes — ${1,2,3,4,5,6}$. You roll $6$ times, so there are $6$ possible outcomes for the first, for each of these there are $6$ possible outcomes for the second roll, and so on. This means there are $6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 6^6$ sequences possible. Only one of these is the sequence you want, so your chance of getting that sequence is $\frac{1}{6^6}$.

Alternatively, you can see it this way. There is a $\frac{1}{6}$ chance of getting a $1$ on your first roll, a $\frac{1}{6}$ chance of getting a $2$ on your second roll, and so on. This means the total probability of getting this particular sequence, is $\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{6^6}$.
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