Problem 6 - Algebra and Trigonometry

Question

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: Determine the possible numbers for the first digit

The first digit of a telephone number cannot be 0 or 1. Therefore, it can be any digit from 2 to 9, which gives us 8 possible choices.

Step 2: Identify the possible numbers for the other six digits

Each of the remaining six digits can be any digit from 0 to 9. Therefore, each digit has 10 possible choices.

Step 3: Calculate the total number of possible combinations

Since the first digit has 8 possible choices and the remaining six digits each have 10 possible choices, the total number of telephone numbers is calculated as follows: \[8 imes 10^6\]Calculate this to find the total number of possible telephone numbers.

Step 4: Compute the result

Compute the expression \(8 imes 10^6\) to find the total number of possible telephone numbers. This results in 8,000,000 possible combinations.

Key Concepts

Telephone Number Combinations

When it comes to telephone number combinations, we are essentially looking at a sequence of digits that forms a unique identifier. Normally, in a simplified setting, we consider a phone number having seven digits. The goal is to determine how these digits can be varied to produce different telephone numbers. This involves some basic principles of combinatorics, specifically focusing on how many ways a sequence can be formed based on certain restrictions or rules. Understanding this concept aids in grasping the big picture of number combinations and their practical applications.

Digit Restrictions

Digit restrictions play a crucial role in determining the possible combinations for telephone numbers. For our exercise, the first digit cannot be 0 or 1. This is a common guideline for many numbering plans to avoid confusion with special service numbers. Thus, the first digit of a telephone number can only be one of the digits from 2 to 9.
This provides us with 8 possible choices for the first position.

• The remaining six digits, however, have no restriction.
• Each can be any digit from 0 to 9.
• This results in 10 possible choices for each of these digits.

This restriction on the first digit is what changes a simple permutation into a slightly more calculated one, as the variety for the initial placement impacts the total results substantially.

Total Combinations Calculation

The calculation of total telephone number combinations is a key part of understanding combinatorics. To determine the total number of possible telephone numbers, you first acknowledge the number of choices available for each digit, considering the given restrictions. From the previous section, the first digit has 8 possibilities.
The remaining six digits have 10 possibilities each.
Now we multiply these options together:

• The multiplication sequence is \[8 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 8 \times 10^6 \]
• Calculating this gives us 8,000,000
• 8,000,000 possible unique telephone numbers exist, respecting our initial restrictions

This simple multiplication helps illustrate how restrictions and combinations work together to form a large number of possibilities, showing the power of combinatorics in real-world situations.