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mathematics, in pictures
Concept

Derangement

An arrangement of a set of objects that leaves none of them in its original place. The proportion of all arrangements that are derangements settles almost immediately on the reciprocal of the number e, because the count is a truncated series for it.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

All 24 arrangements of 4 objects, and the 9 that move every one. Every permutation of 4 objects drawn as a grid of cells, with the diagonal — where an object stays where it began — shaded, and the arrangements that avoid it entirely marked.

Nobody gets their own hat

Hand back a pile of hats at random and ask for the chance that not one person gets their own. The answer barely moves as the crowd grows — it is a third and a bit at four people, and a third and a bit at four thousand.

probability · inclusion exclusion The share of arrangements that fix nothing, up to 8 objects. A bar per number of objects, giving the proportion of its arrangements that leave nothing in place, against the horizontal line at 1/e.

The constant that counts what does not happen

Nothing grows in a shuffled pack of cards, and nothing grows in a factorial. Yet e sits in the middle of both — as the chance that a shuffle leaves nothing in place, and as the base that makes n! nearly a power.

analysis · the exponential

Named alongside it

The objects these essays reach for when they reach for this one.

ApproximationCounting argumente, the numberAlternating seriesComplementary countingCounting two waysFactorialInclusion exclusionLimitPermutationPower seriesProbability

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