Math Problem Statement

How many different seven-digit telephone numbers can be formed if the first digit cannot be zero?

Solution

The question asks:

"How many different seven-digit telephone numbers can be formed if the first digit cannot be zero?"

Solution:

  1. First digit: The first digit cannot be zero, so it can be any of the digits from 1 to 9. This gives us 9 possible choices.

  2. Remaining six digits: Each of these digits can be any number from 0 to 9, giving us 10 choices per digit.

    Therefore, the total number of combinations for these six digits is 10×10×10×10×10×10=10610 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^6.

  3. Total combinations: To find the total number of seven-digit telephone numbers, multiply the choices for the first digit by the combinations of the remaining six digits:

    9×106=9,000,0009 \times 10^6 = 9,000,000

Answer:

There are 9,000,000 different seven-digit telephone numbers that can be formed if the first digit cannot be zero.

Would you like to see more details or have any questions?

Here are some related questions:

  1. How would the total change if the first two digits cannot be zero?
  2. What if the last digit must also be odd?
  3. How many combinations would there be if all digits could be repeated except for the first?
  4. How would the count change if the telephone number must be a palindrome?
  5. What if a digit cannot appear more than twice in the number?

Tip: When dealing with counting problems, always start by breaking down restrictions step-by-step to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Counting Principles

Formulas

Number of combinations = First digit choices * (Remaining digit choices)^6

Theorems

Fundamental Principle of Counting

Suitable Grade Level

Grades 9-12