Question 1025984: if a given set has nine elements, how many of its subsets have at least five elements? Found 3 solutions by richard1234, ikleyn, Edwin McCravy:Answer by richard1234(7193) (Show Source):
You can put this solution on YOUR website! There are 2^9 subsets, and exactly half of them (2^8 = 256) contain at least five elements by symmetry.
If you have a subset with fewer than five elements, then its complement will contain at least five elements.
You can put this solution on YOUR website! .
if a given set has nine elements, how many of its subsets have at least five elements?
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= 1 + 9 + 36 + 84 + 126 + 126 = 382,
where is the number of combinations of n things taken k at a time.
There is = 1 empty subset.
There are = 9 subsets consisting of 1 element.
There are = 36 subsets consisting of 2 elements.
There are = 84 subsets consisting of 3 elements.
There are = 126 subsets consisting of 4 elements.
There are = 126 subsets consisting of 5 elements.
Regarding combinations, see the lesson Introduction to Combinations in this site.
Answer. 382 subsets, including one empty subset.
Notice.
Dear Mr. Edwin McCravy,
in your post you publicly said that my solution is incorrect.
But is IS correct due to obvious reasons.
So I ask you to make a public acknowledgment in your post,
that you take your words back and recognize that you were wrong and my solution is correct.
It doesn't require much time to get the right conclusion.
Finally, it is important for the student to know the truth.
If it is difficult to you to make such contr-statement,
you can simply change your post correspondingly.
When I see it, I will take off this notice from my post.
The first solution above is correct.
But it's not easy to see why half have 5 or more
and the other half don't.
The second solution above is incorrect.
She went for the subsets with fewer than 5 elements,
because their complements all have 5 elements or more.
That way would have been correct. But she made a
mistake and included the subsets with 5 elements, but
she shouldn't have because their complements have only
4 elements.
The correct and the easiest way to understand
is to add the combinations of 9 elements taken
5 or more.
9C5+9C6+9C7+9C8+9C9 = 126+84+36+9+1=256.
Edwin
