In forming three-digit numbers where two digits are the same, how many such numbers can be created with no zeros?

To find the total number of three-digit numbers that can be formed with two identical digits and one different digit, while ensuring that no zeros are involved, we can break down the problem into clear steps.

1. Choosing the Digit That Repeats: Since we are forming a three-digit number with no zeros, the digits can range from 1 to 9. There are 9 possible choices for the digit that will be repeated.

2. Choosing the Different Digit: After selecting the repeating digit, we need to choose a different digit. This digit also must be from the set of 1 to 9 but cannot be the same as the repeating digit. This gives us 8 choices for the different digit.

3. Arranging the Digits: The arrangement of the digits is important as we want to account for different positions where the repeating digit can appear. The possible arrangements of the digits can be calculated by considering that we have two identical digits and one unique digit. The formula to calculate the arrangements in this case is given by:

[

\frac{3!}{2!} = 3

]

This formula accounts for the fact that the two identical digits are indistinguishable