Problem 59 - Algebra and Trigonometry

Question

Subsets A set has eight elements. (a) How many subsets containing five elements does this set have? (b) How many subsets does this set have?

Step-by-Step Solution

Answer

(a) 56 subsets with five elements. (b) 256 total subsets.

Step 1: Understanding Subsets

A subset is a set that contains some or possibly all elements of another set. For any set with \( n \) elements, the number of elements in a subset can range from 0 to \( n \). The total number of subsets of a set is \( 2^n \).

Step 2: Apply Combinatorics for Part (a)

To find the number of subsets containing exactly five elements from a set with eight elements, use the binomial coefficient. This is denoted as \( \binom{n}{k} \), which calculates how many ways you can choose \( k \) elements from \( n \) elements. Here, \( n=8 \) and \( k=5 \).

Step 3: Calculate Binomial Coefficient for Part (a)

The binomial coefficient \( \binom{8}{5} \) is calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Thus,\[\binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56.\]There are 56 subsets of five elements.

Step 4: Apply Subset Formula for Part (b)

The total number of subsets of a set with \( n \) elements is given by \( 2^n \). For a set with eight elements, calculate \( 2^8 \).

Step 5: Calculate Total Subsets for Part (b)

Calculate \( 2^8 = 256 \). This means the set with eight elements has 256 subsets.

Key Concepts

Binomial Coefficient

A binomial coefficient is a fundamental concept in mathematics used to determine how many different ways you can select a certain number of items from a larger set, without regard to the order in which they are selected. Think of it like choosing which team members to pick for a committee out of a much larger group.
The binomial coefficient is often written as \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. The general formula to calculate the binomial coefficient is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Here, \( n! \) (n factorial) means the product of all positive integers up to \( n \).
This formula helps simplify complex selection problems by guiding us on how to calculate combinations efficiently. For instance, if we want to find the number of ways to choose 5 elements from a set of 8, we would use \( \binom{8}{5} \) as discussed in the solution above.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and how they can be counted or arranged. It's a fascinating field that covers topics such as graphs, designs, and arrangements. When thinking of combinatorics, consider any scenario that involves organizing or selecting items.
In this particular exercise, we're using combinatorics to determine subsets. Each subset is a unique group of elements from a larger set. For example, when arranging events or creating groups, combinatorics helps us understand all possible arrangements or selections.
A primary technique within combinatorics is calculating the number of combinations, using the binomial coefficient, which we used in this exercise. Combinatorics not only ensures every possible subset is considered but also teaches efficiency in calculation, especially for larger sets. This is why it's an essential tool in discrete mathematics fields and computer science.

Number of Subsets Formula

The number of subsets formula is another critical concept in set theory, which helps us understand just how many different ways a particular set can be divided. This formula is based on the principle that each element in a set can either be included in or excluded from a subset.
The formula used to calculate the total number of subsets of a set with \( n \) elements is:

• \( 2^n \)

Every element has two states: being in a subset or not. Therefore, if you have a set with 8 elements, the total number of different subsets is calculated as \( 2^8 \), which equals 256.
This formula provides a quick and accurate method to understand how many potential groupings or partitions an entire set of elements can have. It's an easy way to grasp the complexity and vastness of arrangements even in seemingly small sets.