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 Three subsets (Posted on 2014-08-15)
You are requested to create 3 disjoint sets such that:
1. Their union is a set of 10 digits (i.e. integers from 0 to 9 inclusive).
2. The average value of the members of set A is 3.5.
3. The number of members in B is less than the number of members in C.

How many distinct solutions are there?

Rem: No empty sets.

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 solution -- now for real? | Comment 6 of 13 |
(In reply to computer solution by Charlie)

Set A can have two members in any of 4 ways:
{0,7}{1,6}{2,5}{3,4}
In each of these 4 possibilities, B and C can be divided with B getting 1, 2 or 3 out of the 8 remaining.

So far we get 4 * (C(8,1)+C(8,2)+C(8,3))

Set A can have four members in any of 5 ways:
{1,2,3,8}{1,2,4,7}{1,2,5,6}{1,3,4,6}{2,3,4,5}
B can have 1 or 2 of the remaining 6.

So we add in 5 * (C(6,1)+C(6,2))

Set A can have six members in one way:
{0,2,3,4,5,7}
Set B can have only 1 of the remaining four:

When set A has 8 elements, B and C can't be divided unevenly without an empty set.

4 * (C(8,1)+C(8,2)+C(8,3)) + 5 * (C(6,1)+C(6,2)) + C(4,1) = 477

Edited on August 15, 2014, 5:07 pm
 Posted by Charlie on 2014-08-15 17:07:08

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