Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. How many two-letter strings \(-\) the first letter from \(A\) and the second letter from \(B-\) can be formed from the sets \(A=\\{b, c, d\\}\) and \(B=\\{a, e, i, o, u\\} ?\)

Step-by-Step Solution

Answer

15 two-letter strings can be formed.

Step 1: Identify the Principle

To determine how many two-letter strings can be formed from set \(A\) and set \(B\) by selecting one letter from each, we need to decide on the principle to apply. The **Multiplication Principle** is appropriate here because we have two independent choices – one for each letter – requiring us to multiply the number of choices for the first event (selecting a letter from set \(A\)) by the number of choices for the second event (selecting a letter from set \(B\)).

Step 2: Count Possible Choices in Set A

Count the number of elements in set \(A\). Set \(A\) is given as \(\{b, c, d\}\). This set has 3 letters. Thus, there are 3 possible choices for the first letter.

Step 3: Count Possible Choices in Set B

Count the number of elements in set \(B\). Set \(B\) is given as \(\{a, e, i, o, u\}\). This set has 5 letters. So, there are 5 possible choices for the second letter.

Step 4: Apply the Multiplication Principle

Use the Multiplication Principle: multiply the number of choices for the first letter by the number of choices for the second letter. Thus, the total number of two-letter strings is calculated as follows:\[3 \text{ choices from set } A \times 5 \text{ choices from set } B = 15 \text{ total strings}\]

Step 5: Conclusion

Using the Multiplication Principle, there are 15 different two-letter strings that can be formed by taking one letter from set \(A\) and one letter from set \(B\).

Key Concepts

Addition Principle

The Addition Principle is a fundamental concept in combinatorics, particularly useful in deciding between different scenarios. This principle states that if you have two or more mutually exclusive events – meaning they cannot happen simultaneously – the total number of outcomes is found by adding the number of ways each event can occur. Think of it as choosing between alternatives. For example, if you're deciding between 3 types of sandwiches or 4 types of salads, and you can only choose one option, you have a total of 3 + 4 = 7 options.

• It is useful when you have "either/or" situations.
• Ensures no double counting of overlapping choices.

In our exercise, the Addition Principle isn't applicable because we are required to make two consecutive choices, not alternatives, thus multiplying our choices is appropriate instead.

Two-letter strings

Two-letter strings involve creating a sequence of two letters, where each comes from a specific set. In this task, you are forming strings from the sets \(A\) and \(B\). A string in this context does not have to make a meaningful word; it is just an ordered pair of letters.

• Set \(A\) provides the first letter in each string.
• Set \(B\) provides the second letter.

By choosing one letter from \(A\) and another from \(B\), each distinct selection forms a unique two-letter string, such as "ba", "be", etc. This sequence is essential in understanding how to apply the Multiplication Principle for counting.

Sets and elements

Sets are collections of distinct objects, and elements are the individual objects within a set. In mathematics, sets are often used to manage and organize elements systematically.
Consider Set \(A = \{b, c, d\}\) and Set \(B = \{a, e, i, o, u\}\):

• Set \(A\) contains 3 elements: \(b, c, d\).
• Set \(B\) contains 5 elements: \(a, e, i, o, u\).

Knowing the number of elements in sets \(A\) and \(B\) is crucial for applying counting principles effectively. It lays the groundwork for determining combinations and, ultimately, the number of possible outcomes.

Counting principles

Counting principles in mathematics help us determine how many different ways we can arrange or combine objects. Two primary principles guide these calculations: the Addition Principle and the Multiplication Principle.

• Addition Principle: Used when choosing between different options.
• Multiplication Principle: Applied when making consecutive choices, like in our exercise.

The Multiplication Principle states if event 1 can occur in \(m\) ways and event 2 in \(n\) ways, then both events together can occur in \(m \times n\) ways. Therefore, for our two-letter strings, with 3 choices from set \(A\) and 5 from set \(B\), the formula is straightforward:
\[3 \times 5 = 15\]
This shows there are 15 possible two-letter strings that can be formed by these combinations, emphasizing the power of these simple yet effective principles in combinatorics.