Recall that in Section 1.1 we introduced the following arrangement of numbers, which is called Pascal’s triangle.
Notice that the fourth line* of this triangle contains the numbers 1, 4, 6, 4, 1, which, as we saw in Example are precisely the counts of the number of subsets of a four-element set with 0, 1, 2, 3, and 4 elements, respectively.
How do you interpret the sixth line of Pascal’s triangle?
Example Finding All Subsets of a Set Systematically Find all subsets of the set {1, 2, 3, 4}.
We can organize this problem by considering subsets according to their size, going from 0 to 4. This method is illustrated in the following table.
Size of Subset | Subsets of This Size
| Number of Subsets of This Size |
0 | ∅ | 1 |
1 | {1}, {2}, {3}, {4} | 4 |
2 | {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} | 6 |
3 | {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4} | 4 |
4 | {1, 2, 3, 4} | 1 |
5 |
| Total = 16 |
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