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 = {} ?

No it isnt. I am not restricting it to subsets whose union includes all 10 numbers. I am also failing to eliminate duplicates.

Answer: 12,006,914,172 unordered triples, and (12,006,914,172)*(3!) =72,041,485,032 ordered triples. (This is essentially the same as my original answer, 13,680, if you allow for rounding errors on my calculator)

Explanation:

(A int B) int C = {}

If C has just 1 number (there are 10 such sets), then any two sets A and B, where A is any subset of the remaining 9 numbers, and B is any subset of all 10 numbers, should do the trick. A set of 9 elements has (2^9)-1 nonnull subsets. A set of 10 elements has (2^10)-1 nonnull subsets.

If C has 2 number (there are 10*9 such sets), then any two sets A and B, where A is any subset of the remaining 8 numbers, and B is any subset of all 10 numbers, should do the trick. A set of 8 elements has (2^8)-1 nonnull subsets.

etc....

(10)*(2^9-1)*(2^10-1) + (10*9)*(2^8-1)*(2^10-1)
+ (10*9*8)*(2^7-1)*(2^10-1) + (10*9*8*7)*(2^6-1)*(2*10-1)
+ (10*9*8*7*6)*(2^5-1)*(2^10-1)
+ (10*9*8*7*6*5)*(2^4-1)*(2^10-1)
+ (10*9*8*7*6*5*4)*(2^3-1)*(2^10-1)
+ (10*9*8*7*6*5*4*3)*(2^2-1)*(2^10-1)

= 1023*[(10)*(2^9-1) + (10*9)*(2^8-1) + (10*9*8)*(2^7-1)
+ (10*9*8*7)*(2^6-1) + (10*9*8*7*6)*(2^5-1)
+ (10*9*8*7*6*5)*(2^4-1) + (10*9*8*7*6*5*4)*(2^3-1)
+ (10*9*8*7*6*5*4*3)*(2^2-1)]

= 1023*[(10)*(511) + (10*9)*(255) + (10*9*8)*(127)
+ (10*9*8*7)*(63) + (10*9*8*7*6)*(31)
+ (10*9*8*7*6*5)*(15) + (10*9*8*7*6*5*4)*(7)
+ (10*9*8*7*6*5*4*3)*(3)]

= 1023*[(10)*(511) + (90)*(255) + (720)*(127)
+ (5040)*(63) + (30240)*(31)
+ (151200)*(15) + (604800)*(7)
+ (1814400)*(3)]

= 1023*[(5110) + (22950) + (9144)
+ (317520) + (937440)
+ (768000) + (4233600)
+ (5443200)]

= 1023*11736964 = 12,006,914,172


Edited on February 10, 2004, 1:57 pm

Posted by Penny on 2004-02-10 13:02:43