If you're looking for more on this sort of thing, what we did here was "enumerate non-decreasing sequences".

A lot of these elementary enumeration problems hinge on recursion: a non-decreasing sequence of length n+m is comprised of one of length m glued to the end of one of length n.

When we add the (n + 1)-th variable, we need to know how many times the nth variable took on each of its possible values, which can be found in the nth diagonal of Pascal's triangle. (Prove by induction somehow.)

I think that solves the problem.

For the case with just two variables 2 ≤ a ≤ b ≤ 6, there are (7 - a) choices for b once a is fixed, and the sum over a from 2 to 6 of (7 - a) is 15.

Can you do a similar trick for 2 ≤ a ≤ b ≤ c ≤ 6? Can you get a recurrence relation by writing the answer for this case in terms of the previous one?

Are you sure there are only 15?

Does this company only have "AI" in its name for marketing reasons, or is it actually engaged in slop generation?

The plot on the right is exactly what I had in mind!

Can you superimpose the exact and smoothed plots with differences in lighter grey?

Maybe just plotting a smoothed line on top would be more straightforward, I suppose.

Also formalized the frequentist interpretation of probability.

Would the "The Lady Tasting Tea" be too breezy?

Not certain. I don't want a facsimile edition but other than that I'm not too picky.

Maybe I'm too ggplot2-brained, but I wouldn't have guessed that "make one line in this chart dashed and the rest solid" is advanced!

Every time I can't figure out how to get what I want, the solution always seems to be some zany kludge.

I've been on the hunt for a copy of this, but it seems scarcer than his other book, "Graphic Methods for Presenting Facts".

(After live-generating 80 lines of SQL with five paragraphs of pre-written prompts)

"This would have taken hours!"

Maybe for *you*.

Plot by distance to an ocean port and you get a different set of outliers.