The sides of a triangle have lengths of 7, 13, and x. As in

In algebra, it is shown that ; not by coincidence, the set ⭋, which has 0 elements, has 1 subset. Just as 2 1 = 2, the set {a} which has 1 element, has 2 subsets. The pattern continues so that a set with 2 elements has 2 2 ⫽ 4 subsets and a set with 3 elements has 2 3 ⫽ 8 subsets. A quick examination suggests this fact: 2 = 1 Looking back at Example 1, we notice that the number of sub- sets of the four-element set {a, b, c, d} is , which equals 16, or . The preceding principle can be restated in the following equivalent form: 2 4 1 + 4 + 6 + 4 + 1 EXAMPLE 3 The sixth row of Pascal’s Triangle is 1 5 10 10 5 1. Use the principle above to find the sum of the entries of this row. Solution With , it follows that . Then , or 32. 쮿 1 + 5 + 10 + 10 + 5 + 1 = 2 5 n - 1 = 5 n = 6 EXAMPLE 1 List all 16 subsets of the set {a, b, c, d} by considering the fifth row of Pascal’s Triangle, namely 1 4 6 4 1. Notice also that 1 + 4 + 6 + 4 + 1 must equal 16. Solution ⭋, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}. 쮿 EXAMPLE 2 Find the number of subsets for a set with six elements. Solution The number of subsets is 2 6 , or 64. 쮿 NOTE: There are 32 subsets for a set containing five elements; consider {a, b, c, d, e}. In closing, we note that only a few of the principles based upon Pascal’s Triangle have been explored in this Perspective on Application The total number of subsets for a set with n elements is . 2 n The sum of the entries in row n of Pascal’s Triangle is . 2 n - 1 The entries of the fifth row of Pascal’s Triangle correspond to the numbers of subsets of the four-element set {a, b, c, d}; of course, the subsets of {a, b, c, d} must have 0 elements, 1 element each, 2 elements each, 3 elements each, or 4 elements each. Based upon the preceding principle, there will be a total of subsets for {a, b, c, d}. 2 4 = 16 Pascal’s Triangle Set Number of Elements Subsets of the Set Number of Subsets 1 ⭋ ⭋ 1 1 1 {a} 1 ⭋, {a} subsets 1 + 1 2 1 2 1 {a, b} 2 ⭋, {a}, {b}, {a, b} subsets 1 + 2 + 1

4 1

3 3 1 {a, b, c} 3 ⭋, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} subsets 1 + 3 + 3 + 1 8 1 subset of 0 elements, 3 subsets of 1 element each, 3 subsets of 2 elements each, 1 subset of 3 elements