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Non-empty sets element count (Posted on 2015-04-13) Difficulty: 3 of 5
N is the number of ordered pairs of non-empty sets P and Q that have the following properties:
  1. P Q ={1,2,3,4,5,6,7,8,9,10,11,12}, and:
  2. P Q = Φ, and:
  3. The number of elements of P is not an element of P, and:
  4. The number of elements of Q is not an element of Q
Find N.

No Solution Yet Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 4 of 5 |
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
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