What is the probability that you get an increasing sequence of rolls if you roll it three times?

Question

So, I know the answer to this question is: $$\frac{{6 \choose 3}}{6^{3}} = \frac{20}{216} = \frac{5}{54}$$

But I have no intuition.

Why is the numerator ${6 \choose 3}$? Why is the numbers of ways to pick 3 from 6 equal to getting rolls in increasing order? This doesn't make sense to me that it would be that way if theres a condition that it needs to be increasing. Any help to understand this is appreciated!

Answer 1

Assuming by increasing you mean strictly increasing, so ties are not allowed, you have to pick the three rolls to be different, so $6 \choose 3$ is the number of ways to pick the three numbers you roll. Once you have chosen the numbers, there is only one order that is increasing, so there are again $6 \choose 3$ sets of three strictly increasing rolls.