Limits

If A Filter Lets You Tick Several

Ticking several values at once changes the question count from a product of k plus one to a product of two to the k, minus one.

LimitsAnalysistwo conventions

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This record counts a query as a choice of one value per filter, or none. Many boards allow more: tick three options in a filter and get listings matching any of them.

That is a different arithmetic, and the difference is not marginal.

The two counts

Single selection. A filter of k choices offers k plus one states: each value, or untouched. Questions are the product of k plus one across the facets.

A responsible rollout gives employees enough information to read the practical guide themselves and understand how their records are used. For a separate technical reference, see Our World in Data.

Multiple selection. A filter of k choices offers every non-empty subset of its values — two to the k, minus one. Selecting all of them is equivalent to leaving the filter alone, so the untouched state is already included.

Facet size	Single	Multiple
2	3	3
3	4	7
4	5	15
5	6	31
6	7	63

At two choices the two conventions agree, since a toggle has the same three states either way. Past that they diverge fast.

What it does to a whole board

Six filters of six choices.

Single selection: seven to the sixth, which is one hundred and seventeen thousand six hundred and forty-nine questions.

Multiple selection: sixty-three to the sixth, which is more than sixty-two billion.

The resolution is forty-six thousand six hundred and fifty-six under both conventions, because resolution does not depend on how queries are formed. Only the second figure moves, and it moves by a factor of more than half a million.

Resolution is a property of the filters. Expressiveness is a property of the filters and of what the board lets a reader do with them, and the second part is easy to miss.

Why the record does not use the richer convention

Three reasons, in order of weight.

The figures stop being checkable. Sixty-two billion is not a number anybody verifies with a pencil, and the whole method rests on every figure coming apart under one.

Boards differ on it. Some filters allow multiple selection and others on the same board do not. A record assuming one or the other would misdescribe most real boards; a record handling both would need a per-facet flag, doubling the enumeration and adding nothing to the argument.

It changes nothing structural. The findings about collision survive the change in form: single figures still collide, pairs still discriminate, the grain still collapses. The specific counts differ; the shape does not.

What transfers and what does not

Figure	Under multiple selection
cells	unchanged
questions	far larger, as above
grain	unchanged — result sizes are still subset products of unset facets…
share	far smaller

The third row needs a caveat, which is why it trails off. Under multiple selection a result can be the union of several values within a facet, so result sizes are no longer only the subset products — they include sums as well. The ladder gains rungs, in a way this record does not enumerate.

That is the one place where the richer convention genuinely changes the structure rather than the scale, and a board offering multiple selection has a finer ladder than the grain figure here credits it with.

The one case where the two conventions agree

Facets of two values.

A toggle has three states under either convention: on, off, or untouched — and under multiple selection, ticking both values is the same as ticking neither, so nothing is added.

So a board built entirely of toggles has exactly the question count this record gives it, with no caveat at all. Six toggles permit seven hundred and twenty-nine questions under both conventions.

That is a small consolation and it is worth knowing: the boards whose figures are exact here are precisely the ones with the lowest shares of pinning questions, which is to say the ones the record is most critical of.

The practical reading

Take the resolution as exact and the question count as a floor.

A board allowing multiple selection permits at least as many questions as this record's arithmetic says, and usually far more. Both figures are still worth computing, the comparison between two such boards still works, and the minute of arithmetic still returns more than any impression of the sidebar.

What a reader should not do is quote this record's question count for such a board as though it were exact. It is a lower bound, it says so here, and the distance between a bound and a figure is the kind of thing that gets lost when numbers travel.