How Many 2-Digit Even Numbers Can Be Formed Within the Range 3-9 With Repetitions Allowed?

The question of how many 2-digit even numbers can be formed from the even digits between 3 and 9, with or without repetition, is a common type of combinatorics problem. This problem can be approached using basic principles of permutations and combinations.

Understanding the Problem

We are given a range from 3 to 9, which contains five numbers: 3, 4, 6, 7, and 9. Out of these, three numbers (4, 6, and 8) are even. We need to determine the number of 2-digit numbers that can be formed using these digits, with repetition allowed. This means that both the tens and units places of the 2-digit number can be occupied by any of the given digits, including repetitions.

Steps to Solve the Problem

Identify the even digits in the given range: 4, 6, and 8.

Determine the number of choices for the tens place. Since we are allowed to repeat digits, we have 3 choices (4, 6, or 8).

Determine the number of choices for the units place. Again, with repetition allowed, we have 3 choices (4, 6, or 8).

Calculate the total number of 2-digit combinations by multiplying the number of choices for the tens place by the number of choices for the units place. This is a direct application of the basic principle of counting, which states that if one event can occur in (m) ways, and a second event can occur independently of the first in (n) ways, then the two events can occur in (m times n) ways.

Mathematically, the total number of 2-digit even numbers with repetition allowed is given by:

[3 times 3 9]

Example of 2-Digit Even Numbers

Let's list all possible 2-digit even numbers that can be formed by these digits:

44, 46, 48 64, 66, 68 84, 86, 88

Conclusion

In summary, the number of 2-digit even numbers that can be formed from the even digits between 3 and 9, with repetition allowed, is 9. These combinations include:

44, 46, 48 64, 66, 68 84, 86, 88

This problem highlights the application of basic combinatorial principles and is a useful example for understanding the concept of permutations with repetition.