Forming Two-Digit Numbers with Digit Repetition Allowed

To form a two-digit number using the digits from 1 to 9 with repetition allowed, we can break down the problem into simpler steps. This article will explore three different methods to determine the total number of two-digit numbers that can be formed under these conditions. We will also discuss the general approach to forming any natural number from 10 to 99 with and without repetition.

Method 1: Using the Choice Protocol

When forming a two-digit number, the first digit can be any number from 1 to 9. This gives us 9 possible choices. The second digit, also following the rule of repetition allowed, can also be any number from 1 to 9, providing another 9 choices. To find the total number of possible two-digit combinations, we multiply the number of choices for each digit:

9 (choices for the first digit) * 9 (choices for the second digit) 81

Thus, there are 81 two-digit numbers that can be formed using the digits 1 to 9 with repetition allowed.

Method 2: General Natural Number Formula

This method can be applied to any natural numbers from 10 to 99. For these numbers, we can exclude '0' from being the first digit. Therefore, we have 9 possibilities for the first digit (1-9) and 10 possibilities for the second digit (0-9). Using the multiplication principle, we get:

9 (first digit choices) * 10 (second digit choices) 90

Here are some ways to verify this:

Distance Method: Subtract the first number from the last number and add 1 (e.g., 99 - 10 1 90). Combinatorics: Multiply the number of choices for the first digit by the number of choices for the second digit (9 * 10 90). Counting: List all the two-digit numbers from 10 to 99, excluding the combination where '0' is the first digit.

Method 3: Verification with Lists

To further solidify our understanding, let's list some specific examples:

Example 1: Form the numbers using the digits 1 and 9. If repetition is allowed, we have the following combinations: 11, 19, 91, 99 (4 numbers) Example 2: List all possible combinations for the digits 1 and 9, ensuring no leading zero: 11, 19, 91, 99 (4 numbers again)

Note: The terms and conditions exclude numbers with a leading zero, thus, numbers such as 01, 02, 03, etc., are not included in the count.

Conclusion

By utilizing these three methods, we have demonstrated that the total number of two-digit numbers that can be formed from the digits 1 to 9 with digit repetition allowed is 81. This same principle can be applied to form any natural numbers from 10 to 99, ensuring a total of 90 combinations, excluding the leading zero numbers.