A permutation of 
distinct, ordered items in which none of the items is in its
original ordered position is known as a derangement.
If some, but not necessarily all, of the items are not in their original ordered
positions, the configuration can be referred to as a partial derangement (Evans et
al. 2002, p. 385).
Among the 
possible permutations of 
distinct items, examine the number 
of these permutations in which exactly 
items are in their original ordered positions. Then
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is a binomial coefficient and 
is the subfactorial
Here is a table of the number of partial derangements 
for 
, 1, ..., 8:
 |
(4)
|
(OEIS A098825).
