The generalized sum rule
Suppose the tasks T1,T2,…,Tm can be done in n1,n2,…,nm ways,
respectively, and no two of these tasks can be done using the same
way, then there are n1+n2 +…+nm ways to do one of these tasks.
Example. You can choose a project from one of three lists. These lists
contain 4, 5 and 10 possible projects, respectively. Then there are
4+5+10=19 ways to do a project.
! It would not be so if the same project appears in two of the lists.
The sum rule in terms of sets: If A1,A2,…,Am are disjoint sets, then
|A1?A2 ? … ? Am| = |A1|+|A2|+…+|Am|