Problem 20 - Algebra for College Students

Question

If a couple has decided on 6 possible first names for their baby and 5 possible middle names, then how many ways are there for them to name their baby?

Step-by-Step Solution

Answer

30

Step 1 - Understand the Problem

The problem asks for the total number of ways to name a baby given 6 possible first names and 5 possible middle names.

Step 2 - Identify the Choices

There are 6 choices for the first name and 5 choices for the middle name.

Step 3 - Apply the Fundamental Counting Principle

According to the Fundamental Counting Principle, the total number of combinations is the product of the number of choices for each decision. Thus, multiply the number of first names by the number of middle names.

Step 4 - Calculate the Total Number of Combinations

Calculate the total number of ways to name the baby: \[ 6 \times 5 = 30 \]

Step 5 - Interpret the Result

There are 30 different possible ways to name the baby.

Key Concepts

Combinations

To understand how to count combinations, let's start with the basics. Combinations are used when you're trying to figure out how many different groups you can form from a larger set. Here, the parents are choosing a first name and a middle name for their baby from a list.

When calculating combinations, it's important to remember that the order does not matter. In our case, choosing 'Emma' as a first name and 'Grace' as a middle name is different from choosing 'Grace' as a first name and 'Emma' as a middle name. But within the context, the calculation works similarly to combinations because we are only interested in the distinct pairs formed.

The formula for combinations without repetition is \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), but for our baby names example, this will be simplified with basic multiplication.

Decision-Making

Decision-making in mathematics involves choosing options from various sets. In our baby naming problem, we have two decisions to make:

• Choose a first name from 6 options.
• Choose a middle name from 5 options.

By breaking down the problem, it becomes easier to see how each decision impacts the total outcome. Each decision has a certain number of choices, and these choices need to be multiplied to get the total number of ways the baby can be named.
For example, if you choose 'Emma' for the first name, you still have 5 choices for the middle name ('Grace', 'Ava', etc.). This is a fundamental step in solving the problem correctly.

Basic Multiplication

The final concept tying everything together is basic multiplication. According to the Fundamental Counting Principle, the way to find the total number of combinations when dealing with multiple decisions is to multiply the number of choices for each decision.

Here's how it applies to our problem:

• You have 6 choices for the first name.
• For each of these choices, you have 5 choices for the middle name.

So, multiplying these options together gives:

\[6 \times 5 = 30. \]


There you have it. By applying basic multiplication, we find that there are 30 different possible ways to name the baby. Breaking the problem up into steps and understanding each part helps simplify the calculation and ensures accuracy.