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The question in the image asks:

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

### Solution:

A telephone number is made up of **7 digits**. If the first digit cannot be zero, we have the following constraints:

1. The **first digit** can be any number between 1 and 9, so there are **9** possible choices (1-9).
2. The **remaining six digits** can be any number between 0 and 9, so each of these six positions has **10** possible choices.

To find the total number of possible telephone numbers, we multiply the number of choices for each digit:

\[
\text{Total numbers} = 9 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10
\]

This simplifies to:

\[
9 \times 10^6 = 9,000,000
\]

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

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Would you like further details or have any questions?

Here are 5 related questions for practice:
1. How many 7-digit numbers can be formed if the first two digits cannot be zero?
2. If the first digit must be even, how many different seven-digit numbers can be formed?
3. What is the probability of randomly generating a 7-digit number starting with 5?
4. How many distinct 6-digit numbers can be formed if repetition of digits is not allowed?
5. If the second digit also cannot be zero, how does that change the total number of 7-digit numbers?

**Tip**: When solving counting problems, always check if there are any restrictions on the digits or positions to avoid miscalculations.

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

Explore how to calculate the total number of unique seven-digit telephone numbers when the first digit cannot be zero. This guide covers permutations in number theory, with a detailed breakdown of the formula used to arrive at 9,000,000 possible combinations. Suitable for high school students in grades 9 to 12.

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