There are exactly 90 palindrome numbers between 100 and 1000.
Understanding Palindromes in the Given Range
A palindrome is a number that reads the same forwards and backward. The question asks for palindromes between 100 and 1000. This means we are looking for all 3-digit palindrome numbers, as 100 is not a palindrome (it's 100, not 001) and 1000 is a 4-digit number.
For a 3-digit number, let's represent it as ABA, where:
- The first digit is A (hundreds place).
- The second digit is B (tens place).
- The third digit is A (units place).
For the number to be a palindrome, the units' digit must be identical to the hundreds' digit. For example, 121, 545, 999 are all palindromes.
How to Construct a 3-Digit Palindrome
To determine the total count, we analyze the options for each digit based on the rules of palindromes and the 3-digit range:
Digit Selection Breakdown
-
Hundreds' Digit (A):
- Since it's a 3-digit number, the hundreds' digit cannot be zero.
- Therefore, the hundreds' digit (A) can be any number from 1 to 9.
- Number of options for A: 9 (i.e., 1, 2, 3, 4, 5, 6, 7, 8, 9).
-
Tens' Digit (B):
- The tens' digit (B) can be any digit from 0 to 9, as its value does not affect the palindromic property of the number (only the first and last digits need to match).
- Number of options for B: 10 (i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
-
Units' Digit (A):
- As explicitly stated in the reference, "The units' digit can be filled in 1 way since this digit MUST be the same digit as the one in the hundreds' place."
- This means whatever digit was chosen for the hundreds' place, the units' place must match it exactly.
- Number of options for A: 1 (determined by the hundreds' digit).
Calculation
To find the total number of possible 3-digit palindromes, we multiply the number of options for each position:
Number of Palindromes = (Options for Hundreds' Digit) × (Options for Tens' Digit) × (Options for Units' Digit)
According to the provided reference:
Number of palindromes between 100 and 1000 = 9 * 10 * 1 = 90.
This calculation systematically accounts for every possible 3-digit palindrome.
Visualizing the Options
Here's a breakdown in a table format:
| Digit Position | Available Options | Reasoning |
|---|---|---|
| Hundreds' (A) | 9 (1, 2, ..., 9) | Cannot be 0 for a 3-digit number. |
| Tens' (B) | 10 (0, 1, ..., 9) | Can be any digit as it's the middle of the palindrome. |
| Units' (A) | 1 (Same as A) | Must match the hundreds' digit for the number to be a palindrome. |
Examples of 3-Digit Palindromes
Some examples of these 90 palindrome numbers include:
- 101, 111, 121, ..., 191
- 202, 212, 222, ..., 292
- ...
- 909, 919, 929, ..., 999
Each hundreds' digit (1-9) gives rise to 10 unique palindromes (e.g., for hundreds' digit 1, we have 101, 111, ..., 191). Since there are 9 choices for the hundreds' digit, $9 \times 10 = 90$ total palindromes.
