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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)

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solution | Comment 2 of 5 |
assume that one of P,Q is empty,
then the other set has 12 elements and also has the number 12, thus violating either requirements 3 or 4.

So assume P has k elements 1<=k<=11 and thus Q has 12-k elements

now if k=6 then |P|=|Q|=6 and thus 6 can not be a member of P or Q but it has to belong to one of them, thus k is not 6

now for any given k, we have that the element k must belong to Q and 12-k must belong to P.  That gives us k-1 remaining elements to assign to set P.  We have 10 remaining elements to choose these k-1 from, thus this can be done in 10C(k-1) ways and the remaining elements go to Q.

So the total number of combinations is

Sum(10C(k-1), k=1 to 11)-10C5
changing index we get
Sum(10Ck,k=0 to 10)-10C5

Thus there are 772 such ordered pairs

  Posted by Daniel on 2015-04-13 13:46:10
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