Assume that the set S has 10 elements.
How many subsets of S have at most 4 elements?
This question is from the section in my book called "Counting Partitions: Combinations." I would greatly appreciate any help! Thanks!
It seems to me there are 10 ways to pick subsets of 1 element; 10!/(8! 2!)= 45 subsets with 2 elements; 10!/(7!3!) = 120 with three elements and 10!/(6!4!) = 210 subsets with four elements
The total is 385.
Hey, thanks! However, that answer was not right. Any other ideas? You seem to be on the right track... This problem really confuses me.
Ok, the answer is 386! Yay! However, I am not sure why the answer is not 385.... maybe because we had to add c(10,0) into the mix. Thanks so much for the help... I wouldn't have gotten the answer had it not been for your help!
To find the number of subsets of set S with at most 4 elements, we can consider the possible sizes of subsets.
1. Subsets with 0 elements: There is only one such subset, which is the empty set.
2. Subsets with 1 element: Since set S has 10 elements, we can choose any one of them to form a subset. So there are 10 such subsets.
3. Subsets with 2 elements: To choose two elements from a set of 10, we can use the combination formula. The number of ways to choose 2 elements from 10 is given by C(10, 2) = 10! / (2! * (10-2)!) = 45.
4. Subsets with 3 elements: Similarly, the number of ways to choose 3 elements from 10 is given by C(10, 3) = 10! / (3! * (10-3)!) = 120.
5. Subsets with 4 elements: Likewise, the number of ways to choose 4 elements from 10 is given by C(10, 4) = 10! / (4! * (10-4)!) = 210.
Now summing up the counts from all the above cases: 1 (empty set) + 10 (1 element) + 45 (2 elements) + 120 (3 elements) + 210 (4 elements) = 386.
Therefore, there are 386 subsets of set S that have at most 4 elements.