It is a well-known problem to find for n letters the number of the arrangements in which no letter occupies its original place; and the solution of it is given by the following general theorem :— viz., the number of the arrangements which satisfy any r conditions is

where 1 denotes the whole number of arrangements; (1) the number of them which fail in regard to the first condition; (2) the number which fail in regard to the second condition; (12) the number which fail in regard to the first condition, and also in regard to the second condition; and so on: Σ(1) means (1) + (2) … + (r) : Σ(12) means (12) + (13) + (2r) … + (r − 1, r); and so on, up to (12…r), which denotes the number failing in regard to each of the r conditions.