In how many ways can four gifts be exchanged so that no person receives her or his own gift?
Exactly one person can receive his gift in (4,1)D(3) ways. Exactly two people can receive their gifts in (4,2)D(2) ways. So,
Note that I wrote "-1" because there is only one way in which everyone can receive his or her own gift.
A little more information is needed.
In how many ways can four gifts be exchanged
so that no person receives her or his own gift?
Are there four people involved in the exchanging?
Does each person start with one gift?
Can one person to receive more than one gift?
Does an "exchange" mean a reciprocal swap? .
For persons A, B, C and D
If A gets B's gift and B gets A's, then C must get D's, (or C and D get own)
If A gets B's gift and B gets C's, then C must get D's again, (or D will get own)
If A gets B's gift and B gets D's, then C must get A's ( to not get own)
Only these 3 are possible for A, if A gets B's.
If we count the situations for A gets C's...
and the situation for A gets D's...
this will introduce 3+3 more.