D&D GM Needs Help Calculating Stat-Upgrades

Hello! I've never taken a stat class, so bear with me if this seems basic. I have searched for a while for a solution to this, but nothing has been clearly appropriate and reliable, so here I am asking you! Anyway, let's get to the point:

I have the number 25. I am going to roll a D100 (1-100 inclusive) die. If the result of the roll is equal to or below that 25, I will add 1 to it, creating a 26 in its place. I will so this a total of 20 times using whatever the current number is; there is the potential for that number to remain at 25 or reach 45 after all 20 rolls, however unlikely.

What I'm after is a way to calculate the exact probability that, after all 20 rolls, the number will be 25, 26, 27, etc. to 45. I want the probability of each number in the range.

Just in case it's confusing, here's how it would look in practice:

25, passed the roll; it's now a 26. It failed the second roll and remains a 26; it passed the next 3 rolls and became 29, but failed every single roll after, making the final result a 29. What was the probability of ending with this final result?

Anyone here know the solution? I don't expect you to do all of the calculations; I'm perfectly happy with a formula if you can help me get one.

I will also modify how many times we roll and the starting number. If possible, please identify those variables so I can play with them later to calibrate the system! Thanks for taking the time to consider this; I'm dying to hear what you think!
It’s a binomial distribution problem. The probability that the number will remain unchanged on any given throw of the die is 0.75, and 0.25 that it will increase.

You can tabulate the number of “successes” (i.e., adding 1) against the probability where each probability is a single term in the expansion of (0.75+0.25)^20 for 20 throws.
Hmm, on rereading the question, you may be right, which would of course change the question. But that wasn’t entirely clear from what was written.
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