Many impressions

Content and Encounter Are Two Different Counts

A reader meets part of a set, in some order. Divide possible encounters by possible contents and the factorial of what they met is left.

at most 151200 arrangements

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Take a set of statements published about a place of work. Any set: ten sentences, ten pages, ten claims. Call how many there are n.

Nobody meets all of them. A reader arrives somewhere in the middle, gets through some number of them, and leaves. Call that number r. The approach is easier to evaluate when reviewers see the latest product information and document the baseline before any settings are changed.

Two different questions follow, and conflating them is the mistake this record exists to make difficult. Another useful point of reference for the wider question is Gallup Workplace.

Three shapes, three gaps

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; the left-to-right order is the drawing's own and carries no meaning.

Each row is one shape. The strokes are the published statements; the figures are how many ordered encounters are possible and how many distinct contents. The left-to-right arrangement of the strokes is the drawing's own convenience.

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.

Contents and encounters are different sizes

The first question is which statements the reader met. The number of possible answers is the count of r-sized subsets of n, written C(n, r), and this record calls it the contents.

The second question is which, in what order. The number of possible answers there is the count of ordered selections, written P(n, r), and this record calls it the arrangements.

Arrangements is always the larger of the two, and the gap is not small. Eight statements published, four met: seventy contents, one thousand six hundred and eighty arrangements.

The gap is exactly one factorial

Divide arrangements by contents and the answer is r factorial, every time, with no remainder and no exceptions.

The reason is mechanical. Fix a content — a particular set of four statements. How many arrangements correspond to that one content? As many as there are orders of four things, which is twenty-four. The same holds for every content, so arrangements is contents multiplied by twenty-four, and dividing undoes it.

What that quotient measures is worth naming. It is how many different impressions one single content is capable of producing. Four statements met is twenty-four impressions of one content. Six statements met is seven hundred and twenty.

Where those seven hundred and twenty come from

Not from the writing. The statements are fixed; the content is fixed; only the sequence varies, and the sequence is settled by things the publisher does not touch.

Which page a reader entered on. How far down they got before something else took their attention. Which fragment of which sentence a search result put in front of them. Which heading held their eye a half-second longer than the two above it. Whether they arrived by a link somebody sent them, with that person's framing already attached.

Every one of those is a fact about the reader's afternoon. Together they pick one arrangement out of seven hundred and twenty, and the page has no vote.

The publisher chooses which statements exist. The reader's circumstances choose the order. Both halves are necessary and only one of them is anybody's decision.

Why this is not a pessimistic framing

It would be easy to read the factorial as a counsel of despair: if there are seven hundred and twenty readings and you control none of them, why bother writing carefully?

The opposite follows. Because the order cannot be controlled, the only thing that survives every order is a property of the set. A statement that makes sense in isolation survives all seven hundred and twenty arrangements. One that only works after three other statements have landed first survives a fraction of them, and the fraction is computable.

This is the practical content of the record. Not a recommendation about tone or length, but an arithmetic reason that the robustness of a set of statements is a different property from the quality of any one of them, and the two are measured differently.

Reading the shapes in the drawing

Three shapes are drawn. Six statements with three met gives one hundred and twenty arrangements against twenty contents — six impressions per content. Eight with four met gives one thousand six hundred and eighty against seventy — twenty-four per content. Ten with six met gives one hundred and fifty-one thousand two hundred against two hundred and ten — seven hundred and twenty per content.

The arrangements figure grows by a factor of one thousand two hundred and sixty between the first shape and the third, while the number of published statements grows from six to ten. That disproportion is the whole point, and it is why counting what has been published is close to uninformative on its own.

Each stroke in the drawing is one statement. A dashed stroke is one a reader could check from outside; a solid stroke is one they could not; a short pale stroke is one they never reached. That third number has a part to itself further down, and it changes neither figure here: arrangements and contents depend on n and r alone.

The two numbers nobody writes down

A set of pages usually knows how much it has published. That is the only one of the three numbers that is routinely recorded.

How much of it a reader actually meets is rarely measured and frequently assumed to be most of it. How many of the published statements can be checked from outside is almost never counted at all. The arrangements figure needs the second of those; the blind share needs the third; and neither can be inferred from the first.