Assume that the set S has 14 elements.
How many subsets of S have at most 4 elements?
null set --- 1
one element --- C(14,1) = 14
two elements --- C(14,2) = 91
three elements --- C(14,3) = 364
four elements --- C(14,4) = 1001
add them up.
^no, I just tried it and it isn't right. any other suggestions?
To find the number of subsets of set S that have at most 4 elements, we can use the concept of combinations.
First, we need to consider all possible sizes of subsets: 0, 1, 2, 3, and 4.
1. Subsets with 0 elements: There is only one subset with 0 elements, which is the empty set.
2. Subsets with 1 element: There are 14 possible choices for the first element. So, there are 14 subsets with 1 element.
3. Subsets with 2 elements: There are 14 choices for the first element and 13 choices for the second element (excluding the first element). However, since the order of elements in a subset does not matter, we need to divide by 2 to avoid double counting. Therefore, there are (14 * 13) / 2 = 91 subsets with 2 elements.
4. Subsets with 3 elements: Following the same logic as above, there are (14 * 13 * 12) / (3 * 2) = 364 subsets with 3 elements.
5. Subsets with 4 elements: Similarly, there are (14 * 13 * 12 * 11) / (4 * 3 * 2) = 1001 subsets with 4 elements.
Now, we can add up the number of subsets for each of the different sizes:
Number of subsets with ≤4 elements = 1 (0-element subset) + 14 (1-element subsets) + 91 (2-element subsets) + 364 (3-element subsets) + 1001 (4-element subsets)
Number of subsets with ≤4 elements = 1471
Therefore, there are 1471 subsets of set S that have at most 4 elements.
Got it:
C(14,0)+C(14,1)+C(14,3)+C(14,4)=answer
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