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We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...th> function or '''the Euler totient function''', is fundamental in number theory. ...

...f occurrences of the pattern <math>\pi</math> as a substring of the random string <math>X</math>. This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensio ...

:In probability theory, an event <math>A</math> is said to be independent of events <math>B_1,B_2, The idea of the proof is to '''reconstruct''' a random string. ...

...e know this? Since it takes at most <math>d</math> steps to fix any binary string of length <math>d</math> bit-by-bit to any other.) This directly gives us t This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensio ...

...are fairly self-explanatory. In the 1906 4th edition of ''Probability and Theory of Errors'' <ref> ...formatting function, which allows one to convert a fractional number to a string, rounded to a user-specified number of decimal places (the ''precision''). ...

...hed 23 years later. It is a fundamental result in the area of extremal set theory, which studies the maximum (or minimum) possible cardinality of a set syste The idea of the proof is to '''reconstruct''' a random string. ...