There are exactly 90 palindromic numbers between 100 and 999.
Understanding Palindromic Numbers
Palindromic numbers are numbers that remain the same when their digits are reversed. For example, 121 is a palindromic number, as it reads the same forwards and backwards.
Counting Palindromic Numbers from 100 to 999
We can determine how many palindromic numbers exist in the range of 100 to 999 by examining the structure of a three-digit palindrome.
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A three-digit palindrome takes the form of ABA, where A is any digit from 1 to 9 (it cannot be 0, as that would result in a two-digit number) and B is any digit from 0 to 9.
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For the first digit (A), there are 9 possible choices (1 to 9).
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For the second digit (B), there are 10 possible choices (0 to 9).
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The third digit is determined by the first digit and must be the same as A, so it has only 1 choice.
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Therefore, the total number of three-digit palindromic numbers is 9 10 1 = 90.
Detailed Breakdown
Here's a breakdown:
| Digit Position | Choices |
|---|---|
| First Digit (A) | 9 (1 to 9) |
| Second Digit (B) | 10 (0 to 9) |
| Third Digit (A) | 1 (must match the first) |
Total Palindromic Numbers: 9 10 1 = 90
Practical Examples
- 101, 111, 121, ..., 191 are the palindromes starting with '1'.
- 202, 212, 222, ..., 292 are the palindromes starting with '2'.
- And so on, until 909, 919, 929, ..., 999.
Conclusion
As detailed in the reference, there are 90 palindromic numbers between 100 and 999.
