Problem 25 - College Algebra with Modeling and Visualization

Question

How many different 7-digit telephone numbers are possible if the first digit cannot be a 0 or a \(1 ?\)

Step-by-Step Solution

Answer

8,000,000 7-digit telephone numbers are possible.

Step 1: Identify the Restrictions

To determine how many different 7-digit telephone numbers are possible, we first identify the restrictions. Specifically, the first digit cannot be a 0 or a 1.

Step 2: Determine Choices for the First Digit

The first digit cannot be a 0 or a 1, so there are 8 possible choices for the first digit (2 through 9 inclusive).

Step 3: Determine Choices for the Remaining Digits

The remaining six digits can be any digit from 0 to 9, which gives us 10 choices for each of these six positions.

Step 4: Calculate Total Number of Combinations

For the first digit, we have 8 choices. For each of the six remaining digits, we have 10 choices. Therefore, the total number of possible telephone numbers is calculated by multiplying these choices together: \[ 8 \times 10^6 \]

Step 5: Perform the Computation

Multiply the numbers we determined: \[ 8 \times 10^6 = 8,000,000 \] Hence, there are 8 million possible 7-digit telephone numbers.

Key Concepts

Counting Principles

Counting principles form the backbone of combinatorics, providing methods for counting different arrangements or selections systematically. Two main principles help solve such problems: the **addition principle** and the **multiplication principle**. The multiplication principle, sometimes called the "rule of product," is simple yet powerful. It states that if one event can occur in **m** ways and a second event can occur independently in **n** ways, the events together can occur in **m × n** ways. In our example of calculating possible telephone numbers, we apply the multiplication principle. Each digit position represents an event, and we calculate the number of ways for each digit and multiply these values. This principle helps us build larger permutations from smaller options efficiently and ensures accuracy in counting even as complexity increases.

Telephone Number Combinations

Telephone number combinations are common problems in combinatorics that require applying counting principles to compute the number of possible sequences. In essence, a telephone number is a string of digits, each of which can hold different possible values. For a 7-digit telephone number:

• Constraints apply to the overall structure, such as specific rules about the first digit.
• Each remaining digit can typically be chosen with broader flexibility, often from an entire set of numbers.

By approaching the problem step by step, we determine choices for each digit position following any restrictions. Our example shows that by selecting 8 possible choices for the first digit due to restrictions and 10 choices for each subsequent digit, multiplication yields the total number of combinations effectively.

Restrictions in Counting

Restrictions in counting are critical to correctly solving combinatorial problems. They limit which options are available at each step, often significantly altering the final count of possibilities. In this telephone number problem, the restriction is that the first digit cannot be a 0 or 1. This reduces the number of choices for that position to 8 (covering the numbers 2 to 9). By clearly identifying such restrictions, we avoid miscounting possibilities and ensure our solutions are accurate. These restrictions ensure that the solutions are valid within the given problem context. Working through these boundaries highlights the importance of identifying conditions that reduce options, thus simplifying the combinatorial problem.