Equivalence of Definitions of Hat-Check Distribution

Theorem

The following definitions of the concept of Hat-Check Distribution are equivalent:

$X$ has the hat-check distribution with parameter $n$ if and only if:

$\ds \map \Pr {X = k} = \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$

$X$ has the hat-check distribution with parameter $n$ if and only if:

$\ds \map \Pr {X = k} = \dfrac {!k} {n! } \dbinom n k$

where:

$!k$ is the subfactorial of $k$

$\dbinom n k$ is the binomial coefficient.

Beginning with the first definition, we have:

$\ds \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$

$=$

$\ds \dfrac 1 {\paren {n - k }!} \dfrac {n! k!} {n! k!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$

multiplying by $1$

$\ds $

$=$

$\ds \dfrac 1 {\paren {n - k }!} \dfrac {n!} {n! k!} !k$

Definition of Subfactorial

$\ds $

$=$

$\ds \dfrac 1 {n! } \dbinom n k !k$

$\blacksquare$