What is met

Ten Contents, Three Different Pages

The contents figure takes twenty-four distinct values across the record, and the busiest covers twenty-three shapes on unrelated pages.

at most 60 arrangements

Home · What is met · Caution

Ten possible contents. Which page is this?

Five published with two met gives ten contents. Five published with three met gives ten. Ten published with one met gives ten. Three pages, one figure, and the third of them is a different size of set entirely.

One contents figure, three pages

dashed: a statement a reader can check from outside
solid: published, but nothing to check it against
short and pale: never reached by this reader

Three shapes drawn; all of them give the same 10 different contents; the left-to-right order is the drawing's own and carries no meaning.

All three rows give ten contents. The published counts are five, five and ten; the arrangements figures are twenty, sixty and ten.

Computed from the three numbers alone. How many statements exist, how many are met and how many can be checked is a stated convention; the counts are arrangements and combinations of whole numbers.

Twenty-four values, three hundred and thirty-eight shapes

Contents takes twenty-four distinct values anywhere in this record. Arrangements, which was already shown to be badly crowded, takes thirty-five.

So the contents column is the more crowded of the two, and by a noticeable margin. Twenty-three shapes carry the value ten. Twenty-five carry the value one. Twenty-two carry two hundred and ten. Operational context is most useful when managers explore the productivity toolkit and read the output beside the work product rather than on its own.

The crowding has two separate sources and they compound. One is the mirror: within a page, r met and n − r met give the same figure, so every page contributes its values in pairs. The other is collision across pages, which has no tidy structure at all — ten arises from C(5, 2), from C(5, 3) and from C(10, 1), and those three have nothing in common beyond the answer. A broader source for checking the surrounding technical or organisational context is Our World in Data.

Why a small set and a large one meet here

C(10, 1) is ten because there are ten statements and the reader met one of them. C(5, 2) is ten because there are ten pairs available in five statements.

Both are ten and the situations are not comparable. In the first, the reader's whole account of the set is a single statement, and nine tenths of what was published is absent. In the second, the reader met two fifths of the set and holds a pair.

A report that gives only the contents figure cannot distinguish a reader who met a tenth of a long set from one who met two fifths of a short one. Those are the two ends of what this record is about.

A figure that cannot tell a tenth of ten from two fifths of five is not reporting on the reader. It is reporting on a count of combinations, which is a different subject.

The arrangements column does separate them

Twenty, sixty, ten. The three rows of the drawing have distinct arrangements figures even though their contents figures agree.

This is the complementary case to the one shown earlier in the record, where three pages shared an arrangements figure of one hundred and twenty and were told apart by contents. Each column rescues the other in different places, and neither does it everywhere.

Together they never fail, which was established in the first part of this record and proved rather than observed. The crowding in one column and the crowding in the other do not overlap at any point in the table.

What this means for reading a single figure anywhere

The record's position, stated once and applied throughout, is that no derived figure travels alone. Every drawing carries the shape — how much published, how much met, how much checkable — in the same row as the figures, and the prose names the quantity each time.

That convention is tedious and it is what makes the collisions above harmless. A reader of this record always knows which of the twenty-four values they are looking at and which page produced it.

Why arrangements is the less crowded column

Thirty-five values against twenty-four, over the same three hundred and thirty-eight shapes. The difference has one cause.

Contents is symmetric within a page, so every page contributes its values in matched pairs and the number of distinct values it can contribute is cut accordingly. Arrangements has no such symmetry: P(n, r) rises strictly with r, so the six reading depths of a ten-published page give six distinct arrangements figures and only five distinct contents figures, four met and six met having collided. On a seven-published page the six depths give six arrangements figures and three contents figures.

Arrangements collides only across pages, by coincidence of products. Contents collides across pages and within every page, by construction. That is the whole of why one column has eleven more values than the other, and it is the reason the two together never fail to identify a page.

The contents figures that are not crowded

Nine of the twenty-four values come from one page each: two, eight, nine, twenty, thirty-six, forty-five, seventy, one hundred and twenty, and two hundred and fifty-two. For those nine, the contents figure alone does identify the page.

The largest of them is two hundred and fifty-two, which is C(10, 5), the peak of the largest page in the record. No other page reaches it, and the reason is simply that the table stops at ten published: a second page giving two hundred and fifty-two contents would need eleven or more.

That is the right way to hold all nine of these. They are uncrowded inside this table and most of them would not stay uncrowded in a larger one. Whether a figure identifies a page is a property of where the record's ranges were drawn, not a property of the arithmetic, and the two are easy to confuse because they look the same on the page.

Twenty-three shapes at ten contents and eleven at two hundred and fifty-two is a distribution of the record's own construction. Nothing in the world is shaped like that, and nothing here claims otherwise.