Determine with proof the number of ordered triples of sets which have the property (i) and (ii) . So basically this means that all the sets are disjoint.

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- Jul 25th 2007, 05:28 PMtukeywilliamsCounting Sets
Determine with proof the number of ordered triples of sets which have the property (i) and (ii) . So basically this means that all the sets are disjoint.

- Jul 25th 2007, 06:55 PMThePerfectHackerQuote:

So basically this means that all the sets are disjoint.

Think of a Venn diagram, see below.

Now the numbers can be placed anyway**except**into the shaded area. So we are placing "marbles" into 6 "boxes" (because there are 7 sections and we omit the middle section). The formula is:

- Jul 25th 2007, 07:05 PMtukeywilliams
Could you also use matrices to solve this problem?

- Jul 25th 2007, 08:02 PMThePerfectHacker
I really have no idea? I am not a Combinatorics expert :(.

Perhaps, you can phrase this problem as a Graph Theory problem and then arrive at your answer by computing the adjancency matrix. But I have no idea how to do that. - Jul 25th 2007, 08:07 PMtukeywilliams
I was thinking that we could somehow use a binary relation. So there is some type of mapping (maybe a bijection?) between , and some type of matrix with so that there are no rows that are or .

- Jul 25th 2007, 09:52 PMtukeywilliams
I think there are possibilities for each row and therefore total possible triples?

- Jul 26th 2007, 04:34 AMPlato
On a strict reading of this problem as stated, I agree that the answer is .

However, are you sure that you have stated this question correctly?

As stated, the question has several different readings.

I have seen this sort of question many times: “How many ways can {0,1,2,…,8,9} be partitioned into three non-empty, pair-wise disjoint sets?”

Could that be what was meant by this question?