Assume that the set S has 7 elements.
How many subsets of S have at most 3 elements
The total number of subsets = 2^7 = 128
This includes the null set
So what we want is
the null set, the set of singles, the set of doubles and the set of triples
= 1 + C(7,1) + C(7,2) + c(7,3)
= 1 + 7 + 21 + 35
= 64
So the number of subsets with at most 3 elements is 64
Well, let's count the subsets, shall we?
First, we have the empty set with 0 elements.
Next, we have subsets with 1 element. Since there are 7 elements in the set S, that means there are 7 subsets with 1 element.
Moving on, we have subsets with 2 elements. To find out how many there are, we can use the binomial coefficient formula, also known as "n choose k." In this case, n is 7 and k is 2. So, we have 7 choose 2, which is equal to 21. That means there are 21 subsets with 2 elements.
Lastly, we have subsets with 3 elements. Applying the same formula, we have 7 choose 3, which equals 35. So, we have 35 subsets with 3 elements.
To find out how many subsets have at most 3 elements, we simply add up the number of subsets with 0, 1, 2, and 3 elements.
0 + 7 + 21 + 35 = 63
Therefore, there are 63 subsets of set S that have at most 3 elements.
To find the number of subsets of set S with at most 3 elements, we need to consider the number of subsets with 0, 1, 2, and 3 elements and add them up.
1. Number of subsets with 0 elements:
There is only one subset with 0 elements, which is the empty set {}. We can consider it as choosing 0 elements from S.
So, there is 1 subset with 0 elements.
2. Number of subsets with 1 element:
For each element in S, we can either choose it or not choose it.
So, the number of subsets with 1 element is equal to the number of elements in S, which is 7.
3. Number of subsets with 2 elements:
To find the number of subsets with 2 elements, we can choose any 2 elements from S.
This can be determined by the combination formula: nCr = n! / (r! * (n-r)!)
The number of subsets with 2 elements in S is 7C2 = 21.
4. Number of subsets with 3 elements:
Similarly, using the combination formula, we can choose any 3 elements from S.
The number of subsets with 3 elements in S is 7C3 = 35.
To find the total number of subsets with at most 3 elements, we sum up the above results:
1 subset with 0 elements + 7 subsets with 1 element + 21 subsets with 2 elements + 35 subsets with 3 elements = 1 + 7 + 21 + 35 = 64.
Therefore, there are 64 subsets of S that have at most 3 elements.
To find the number of subsets of set S with at most 3 elements, we need to consider all possible sizes of subsets: 0, 1, 2, and 3 elements.
1. Number of subsets with 0 elements: There is only one subset with 0 elements, and that is the empty set.
2. Number of subsets with 1 element: Since set S has 7 elements, there are 7 possible choices for a 1-element subset. Therefore, there are 7 subsets with 1 element.
3. Number of subsets with 2 elements: To find the number of subsets with 2 elements, we need to choose 2 elements from the 7 elements in set S. This can be calculated using combinations, denoted as C(n, r), which represents the number of ways to choose r elements from a set of n elements.
In this case, we need to calculate C(7, 2).
C(7, 2) = 7! / (2! * (7-2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21.
Therefore, there are 21 subsets with 2 elements.
4. Number of subsets with 3 elements: Using the same logic as above, we need to calculate C(7, 3).
C(7, 3) = 7! / (3! * (7-3)!) = 7! / (3! * 4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35.
So, there are 35 subsets with 3 elements.
To find the total number of subsets with at most 3 elements, we simply sum up the subsets of each size:
Total = Number of subsets with 0 elements + Number of subsets with 1 element + Number of subsets with 2 elements + Number of subsets with 3 elements
= 1 + 7 + 21 + 35
= 64.
Therefore, there are a total of 64 subsets of set S that have at most 3 elements.