Problem 8 - Algebra and Trigonometry

Question

Phone Numbers Telephone numbers consist of seven digits; the first digit cannot be 0 or 1. How many telephone numbers are possible?

Step-by-Step Solution

Answer

There are 8,000,000 possible telephone numbers.

Step 1: Understanding the Problem

We need to compute the total number of possible seven-digit telephone numbers, given the restriction on the first digit.

Step 2: Counting the Choices for the First Digit

The first digit of the telephone number cannot be 0 or 1, so it can be any digit from 2 to 9. That's a total of 8 possible choices for the first digit.

Step 3: Counting the Choices for the Remaining Digits

The remaining six digits have no restrictions, meaning each of these digits can be any digit from 0 to 9. Hence, there are 10 possible choices for each of these six digits.

Step 4: Calculating the Total Number of Telephone Numbers

We calculate the total number of telephone numbers by multiplying the number of choices for each digit. This gives us: \[ 8 \times 10^6 \], since there are 8 choices for the first digit and 10 choices for each of the six subsequent digits.

Step 5: Final Calculation

Compute the final number: \[ 8 \times 10^6 = 8,000,000 \].

Key Concepts

Permutations

In combinatorics, permutations are all about arranging objects in a specific order. When dealing with permutations, every possible order of the arrangement is considered unique. In the telephone number problem, though permutations might not be directly applied since each digit doesn't change positions, understanding permutations helps in realizing how choices are being arranged.
For example, if you have three digits and want to create unique three-digit sequences, permutation rules apply. But in our specific problem of phone numbers, we're mainly concerned with selecting the number of options for each position in the phone number rather than rearranging them.

Counting Principles

Counting principles are the backbone of solving combinatorial problems like the telephone numbers problem. The fundamental counting principle states that if you have multiple stages of decisions, the total number of outcomes is the product of the number of choices at each stage.
This principle is used extensively in the telephone number problem. For the seven-digit number:

• There are 8 options for the first digit (since it can't be 0 or 1).
• Each of the next six digits can be any of the 10 digits from 0 to 9.

By multiplying the number of choices for each position, we get the total number of possible seven-digit phone numbers: \[8 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 8 \times 10^6 = 8,000,000\].
Understanding these principles simplifies the process of counting possible configurations in such problems.

Mathematics Problem-Solving

Mathematics problem-solving often begins with understanding the problem, a crucial step in any solution process. In our telephone number problem, understanding the restrictions such as the first digit not being 0 or 1 is essential. Problem-solving in mathematics then follows with logical steps to reach a solution.
These steps include:

• Breaking down the problem: Here, we considered the telephone number as a series of 7 independent digits, with specific restrictions on the first.
• Applying counting principles: We used the fundamental counting principle to multiply possibilities for each digit.
• Performing calculations: Finally, calculating \[8 \times 10^6\] gave us the solution.

Effective use of these problem-solving strategies leads to a better understanding and quicker solutions to complex problems.