Question

How many integers between 1 and 1000 do not contain repeated digits?

Short answer

The total number of integers between 1 and 1000 without repeated digits is 9 + 81 + 648 = 738.

Step-by-step solution

- Identify Total Numbers

First, recognize that we need to find all the integers between 1 and 1000. Hence, we are looking at numbers from 1 to 999 as well as the number 1000.

- Consider Special Case

Among the numbers listed, the number 1000 is a special case because it does contain repeated digits (three zeros). Thus, it does not meet our criteria and should be excluded.

- Count 1-digit Numbers

The one-digit numbers are from 1 to 9. Each of these numbers obviously does not have any repeated digits. So there are 9 such numbers.

- Count 2-digit Numbers

For two-digit numbers (from 10 to 99), the tens place has 9 options (1 through 9), and the units place has 9 remaining options (0 through 9, excluding the tens place digit). This gives us a calculation of 9 * 9 = 81 two-digit numbers without repeated digits.

- Count 3-digit Numbers

For three-digit numbers (from 100 to 999), we have: The hundreds place has 9 choices (1 through 9), the tens place has 9 choices (0 through 9, but excluding the hundreds place digit), and the units place has 8 choices (0 through 9, excluding the digits in both the hundreds and tens places). This results in 9 * 9 * 8 = 648 three-digit numbers without repeated digits.

Key concepts

Unique Digit Numbers

Unique digit numbers are numbers in which no digit repeats.
For example, the number 123 is a unique digit number because none of the digits 1, 2, or 3 repeats.
On the other hand, the number 122 is NOT a unique digit number because the digit 2 repeats.
When counting unique digit numbers, it's important to carefully choose from the available digits to avoid any repetitions.

Let's explore how to find unique digit numbers within a specific range.

Combinatorial Counting

Combinatorial counting helps us systematically count the number of ways to choose items from a set.
We use this method to solve many types of problems, including our problem of finding numbers without repeated digits.
For example, when looking at the number of different ways we can choose the digits for a two-digit number without repeating any digits, we start by choosing the tens place.
That gives us 9 options (1 through 9), because a number can't start with zero.
Then, for the units place, we have 9 choices remaining (which includes 0 but excludes the digit already chosen for the tens place).
Therefore, the number of unique digit two-digit numbers is 9 options for tens times 9 options for units: \(9 \times 9 = 81\).

Combinatorial counting often involves multiplication principles, where the choices available for each decision step are multiplied together.

Integer Range

The integer range in this problem includes all numbers between 1 and 1000.
It's helpful to break this range down into manageable parts, such as 1-digit, 2-digit, and 3-digit numbers.
Here’s how we divide the problem:

• 1-digit numbers: From 1 to 9
• 2-digit numbers: From 10 to 99
• 3-digit numbers: From 100 to 999
• 1000 is also considered, but it does not have unique digits as it contains three zeros

Understanding this breakdown helps in organizing our counting process.
By carefully analyzing each subset, we ensure all possibilities are counted accurately.
Each category sustains a specific approach for counting unique digit numbers based on its own constraints and combinations.