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Set Me Up (Posted on 2004-02-10) Difficulty: 3 of 5
To demonstrate set union and intersection to her class, Mrs. Putnam asked for three students to each write down a set of numbers.

After they had done so, she looked at their sets and told the class, "the union of these three sets is the first ten counting numbers, but their intersection is empty!"

How many triples (A, B, C) of sets are there such that

A U B U C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
and
A ∩ B ∩ C = {} ?

See The Solution Submitted by DJ    
Rating: 4.3636 (11 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution my solution | Comment 2 of 31 |
Answer: =58968

Why:
Each of the numbers 1,2,3,....10 belongs to one of the sets- we assume no empty set due to the teachers demand.
Let ABBBCACCCB denote situation where 1 and 6 are in A , 2 3 4 AND 10 in B etc
Clearly it isomorphic to a base- THREE 10 digit presentation where all 3 digits have to be present

all possibilities: 3^10
two digits only 3* (2^10-2)
one digit only 3

Answer: 3^10-3* 2^10 - 6-3 =59049- 3*1022 - 3=58968


ady

Edited on February 10, 2004, 10:44 am
  Posted by Ady TZIDON on 2004-02-10 10:33:08
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