Problem 22 - Algebra for College Students

Question

How many three-element subsets containing one vowel and two consonants can be formed from the set \(\mid a, b, c\), d, e, f, g, h, i\\}? 45

Step-by-Step Solution

Answer

There are 45 subsets with one vowel and two consonants.

Step 1: Identify Vowels and Consonants

The set given is \(\{a, b, c, d, e, f, g, h, i\}\). The vowels in this set are \(\{a, e, i\}\) and the consonants are \(\{b, c, d, f, g, h\}\).

Step 2: Choose One Vowel

We can choose one vowel from the set of vowels \(\{a, e, i\}\). The number of ways to do this is \(\binom{3}{1} = 3\).

Step 3: Choose Two Consonants

We can choose two consonants from the set of consonants \(\{b, c, d, f, g, h\}\). The number of ways to choose two consonants is \(\binom{6}{2} = 15\).

Step 4: Calculate Total Number of Three-element Subsets

The total number of possible three-element subsets that include one vowel and two consonants is the product of the choices for vowels and consonants: \(3 \times 15 = 45\).

Key Concepts

Understanding Subsets

A subset is a selection of elements from a larger set. In other words, each subset contains elements that are all part of the original set. When we form subsets, we need to decide how many elements we want in each subset. For example, with the set \(\{a, b, c, d, e, f, g, h, i\}\), each subset can include any number of the set's elements—from none at all to all of them.

Subsets can be of various sizes and the number of subsets formed from an original set depends on the number of elements that the subset is supposed to have. For instance, if you want to create three-element subsets, you would select exactly three elements for each subset.

This exercise requires you to form subsets of exactly three elements from a larger set. It adds a twist by limiting the selection: only one element can be a vowel and the other two must be consonants.

Identifying Vowels and Consonants

Recognizing vowels and consonants in a set is essential for solving this type of problem. In the English language, the vowels are typically

• a
• e
• i
• o
• u

For the given set, \(\{a, b, c, d, e, f, g, h, i\}\), the vowels are \(\{a, e, i\}\). The remaining letters are consonants: \(\{b, c, d, f, g, h\}\). Vowels and consonants behave distinctively in language, and here, the differentiation allows you to create specific types of subsets.

By identifying which letters are vowels and which are consonants, you are well-equipped to tackle the problem of forming a mixture of vowels and consonants within a subset formation.

Using Combinations

The concept of combinations is used to find the number of ways to choose items from a group, where order doesn't matter. When creating subsets, we often use combinations to figure out how many ways you can select the required number of items from each category.

The formula for combinations is given by \(\binom{n}{r}\), where \(n\) is the total number of elements to choose from, and \(r\) is the number of elements you want to choose. This ensures each combination is counted only once, regardless of order.

For the problem we are handling, first, you choose one from the three vowels: \(\binom{3}{1} = 3\). Next, you select two consonants from the set of six: \(\binom{6}{2} = 15\). Finally, multiply these two results together to find the total number of subsets that can be formed:

• Vowel choices: 3 ways
• Consonant choices: 15 ways

Thus, the total number of three-element subsets containing one vowel and two consonants is \(3 \times 15 = 45\).