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Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
ISBN: 9781305965584
BuyFindarrow_forward

Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
ISBN: 9781305965584
Textbook Problem

Subsets and Pascal’s Triangle Following is a list of all the subsets of (a, b, c, d).

Subsets with

0 elements: { }

1 element: {a}, {b}, {c}, {d}

2 elements: {a, h}, {a, c}, {a, d}, {h, c}, {b, d}, {c, d}

3 elements: {a, h, c}, {a., b, d}, {a, c, d}, {b, c, d}

4 elements: {a, b, c, d}

There is 1 subset with zero elements, and there are 4 subsets with exactly one element, 6 subsets with exactly two element, 4 subsets with exactly three elements, and I subset with exactly four elements. Note that the numbers 1,4,6,4, 1 are the numbers in row 4 of Pascal’s triangle, which is shown in the column to the right. Recall that the numbers in Pascal’s triangle are created in the following manner. Each row begins and ends with the number I. Any other number in a row is the sum of the two closest numbers above it. For instance, the first 10 in row 5 is the sum of the first 4 and the 6in row 4.

a. Use Pascal’s triangle to make a conjecture about the numbers of subsets of {a, b, C, d, e} that have: item elements, exactly one element, exactly two elements, exactly three elements, exactly four elements, and exactly five elements. Use your work from Check Your Progress 4 on page 60 to verify that your conjucture is correct.

b. Extend Pascal’s triangle to show row 6. Use row 6 of Pascal’s triangle to make a conjecture about the number of subsets of {a, b, c, d, e, f} that have exactly three elements. Make a list of all the subsets of (a, b, c, d, e, f) that have exactly three elements to verify that your conjecture is correct.

a.

To determine

To find the number of subsets that have zero elements, exactly one element, exactly two elements, exactly three elements, exactly four elements, and exactly five elements and to verify the same by finding the subsets.

Explanation

Given information:

Given, the number of subsets with the number of elements in the subset follows the pattern of the Pascal’s triangle.

  

The set being considered is {a, b, c, d, e}

Calculation:

The given set consists of 5 elements.

Hence, the number of subsets having with a given number of elements follow the Pascal triangle row 5.

Therefore,

The number of subsets with zero elements is 1.

The number of subsets with one element is 5.

The number of subsets with two elements is 10.

The number of subsets with three elements is 10.

The number of subsets with four elements is 5.

The number of subsets with five elements is 1.

This can be verified by writing the subsets.

The subsets with zero elements is given by: {}

Therefore, the number of subsets with zero elements is 1.

The subsets with one element are given by: {a},{b},{c},{d},{e}

Therefore, the number of subsets with one element is 5.

The subsets with two elements are given by: {a, b},{a, c},{a, d},{a, e},{b, c},{b, d},{b, e},{c, d},{c, e},{d, e}

Therefore, the number of subsets with two elements is 10

b.

To determine

To find the row 6 of Pascal’s triangle. To find the number of subsets of the set {a, b, c, d, e, f} that have exactly three elements and to verify it by writing all the subsets having exactly three elements.

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