A Pre-RMO question! -22
If the number of subsets of { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } that are subsets of neither { 1 , 2 , 3 , 4 , 5 } nor { 4 , 5 , 6 , 7 , 8 , 9 } is k , find the sum of the digits of k .
The answer is 6.
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2 solutions
I thought of the exact same thing but wasn't sure if I was right. Thanks!
@Zakir Husain , @X X , @Chew-Seong Cheong , the two solutions given here have different answers, I can't understand which one is right. Please help! Thanks!
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I think this one, the answer might be 420
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I got only 418. I think only 2 subsets should be removed, not 4. Can you please tell my mistake?
A = ( 1 , 2 , 3 , 4 , 5 ) B = ( 4 , 5 , 6 , 7 , 8 , 9 )
To form a set that is neither a subet of A nor B, we need the set C
- with some non-empty blue elements ( ( 1 3 ) + ( 2 3 ) + ( 3 3 ) = 2 3 − 1 = 7 )
- with some non-emplty red elements ( ( 1 4 ) + ( 2 4 ) + ( 3 4 ) + ( 4 4 ) = 2 4 − 1 = 1 5 )
- if we have both blue and red elements, then we can have any purple elements ( ( 0 2 ) + ( 1 2 ) + ( 2 2 ) = 2 2 = 4 )
The total number of set C satisfies the requirement = 7 ∗ 1 5 ∗ 4 = 4 2 0
There are 2 element in the intersection of {1,2,3,4,5} and {4,5,6,7,8,9}.
We can use the idea of Venn's Diagram and get 2 9 − 2 5 − 2 6 + 2 2 = 4 2 0 .