Intuitive Understanding of this probability drawings question I'm doing this problem from Carol Ash's The Probability Tutoring Book

Draw 20 times from the integers {1,2,3...,100}. Find the probability that your draws come out in increasing order if the drawings are with replacement

My attempt:
We are drawing with replacement, the probability of getting a greater number the 2nd, 3rd, 4th and 5th time depend on the previous draws, so I reasoned:
$$\sum_i\frac{1}{100^4}{\frac{100 - i}{100}}$$
But this doesn't work because $i$ - the variable we sum over - isn't measuring what our greatest number at any time is.
So I peeked at the answers:
$$\frac{100 \choose 20}{100^{20}}$$
The denominator, as I understand it, is the total number of possible drawings with replacement. The numerator is the total number of groups of 20 that I can make from the 100 numbers.
But these groupings don't have to necessarily be in increasing order. Also, I learned in high school that when we use the combination operator we draw without replacement... Can someone explain the numerator to me please(whether mathematically or intuitively)?
 A: In order for the draws to be in increasing order, the numbers drawn must all be distinct, so you must have drawn a set of $20$ different numbers. For each possible set of $20$ distinct integers there is exactly one string of draws in which the numbers are increasing, so there is exactly one successful (i.e., increasing) string of draws for each $20$-element subset of $\{1,2,\ldots,100\}$. There are $100^{20}$ possible strings of draws, so the probability of getting a successful one is $$\frac{\binom{100}{20}}{100^{20}}\;$$
The rule of thumb that we count combinations when the draw is without replacement and permutation when it is with replacement applies to straightforward problems, but it can’t be applied blindly in more involved settings. Here the point is that each draw in which the numbers are increasing is completely determined by the set of $20$ numbers drawn, because there’s exactly one increasing draw for each set of $20$ numbers. Thus, counting the sets of $20$ numbers actually amounts to counting the increasing draws of $20$ numbers: there’s a bijection between sets of $20$ numbers and increasing draws of $20$ numbers.
The real distinction is that combinations count sets of things of a certain size, while permutations count sequences of things.
