N is the number of ordered pairs of non-empty sets P and Q that have the following properties:
- P ⋃ Q ={1,2,3,4,5,6,7,8,9,10,11,12}, and:
- P ⋂ Q = Φ, and:
- The number of elements of P is not an element of P, and:
- The number of elements of Q is not an element of Q
Find N.
P has cardinality 1 and Q 11: P = {11}, count = 1
P has cardinality 2: P contains 10 but not 2, plus one of the other 10 numbers: count = C(10,1) = 10
P has cordinality 3: P contains 9 but not 3, plus two of the other ten: count = C(10,2) = 45.
P has cardinality 4: count = C(10,3) = 120.
P has cardinality 5: count = C(10,4) = 210.
Cardinality 6 is impossible (can't both contain and not contain 6).
Symmetric results hold for Q.
1+10+45+120+210=386
Double for the cardinality of Q being the above: 2*386 = 772.
|
|
Posted by Charlie
on 2015-04-13 14:22:49 |
solution
