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!
assistance needed
To find the number of subsets of set S with at most 4 elements, we can consider all possible numbers of elements in the subsets: 0, 1, 2, 3, and 4.
If we have 0 elements in the subset, there is only one possibility, the empty subset.
If we have 1 element in the subset, we can choose any one of the 10 elements from set S. So, the number of subsets with 1 element is 10.
If we have 2 elements in the subset, we can choose any 2 elements from the 10 elements in set S. The number of ways to choose 2 elements out of 10 is given by the combination formula nCr, which is calculated as n! / (r!(n-r)!), where n is the total number of elements, and r is the number of elements to be chosen. Therefore, the number of subsets with 2 elements is 10C2 = 45.
Similarly, if we have 3 elements in the subset, we have 10C3 = 120 possible subsets.
If we have 4 elements in the subset, we have 10C4 = 210 possible subsets.
To find the total number of subsets with at most 4 elements, we add up the subsets from each case: 1 + 45 + 120 + 210 = 376.
So, there are 376 subsets of set S with at most 4 elements.