There are 99 palindromes between 10 and 100,000, including 11 and 999. If we exclude 11 and 999, the answer is 97 palindromes.
Understanding Palindromes
A palindrome is a number (or word) that reads the same backward as forward. For instance, 121, 353, and 9119 are palindromes. We need to count all such numbers within the specified range.
Counting Palindromes Between 10 and 100000
We can systematically count the palindromes by considering the number of digits they have:
Two-Digit Palindromes
- These have the form "AA" where A is a digit from 1 to 9. The only two-digit palindrome between 10 and 100000 is 11 (22,33...99 also count), as stated in the reference.
- There are 9 such palindromes (11, 22, 33, 44, 55, 66, 77, 88, 99).
Three-Digit Palindromes
- These have the form "ABA" where A can be a digit from 1 to 9 and B can be a digit from 0 to 9.
- There are 9 choices for A and 10 choices for B, giving 9 * 10 = 90 three-digit palindromes. Examples include 101, 111, 121,..., 999.
Four-Digit Palindromes
- These have the form "ABBA" where A can be a digit from 1 to 9 and B can be a digit from 0 to 9.
- Similar to the three-digit palindromes, there are 9 choices for A and 10 choices for B, thus resulting in 9 * 10 = 90 four-digit palindromes. Examples include 1001, 1111, 1221, ..., 9999.
Five-Digit Palindromes
- These have the form "ABCBA" where A can be a digit from 1 to 9, and B and C can be digits from 0 to 9.
- There are 9 choices for A, and 10 choices for B and C each, resulting in 9 10 10 = 900 five-digit palindromes. Examples include 10001, 10101, 10201, ..., 99999.
Calculating the Total
To get the total count, we add up the number of palindromes from each category:
- Two-digit palindromes: 9
- Three-digit palindromes: 90
- Four-digit palindromes: 90
- Five-digit palindromes: 900
Total = 9 + 90 + 90 + 900 = 1089
However, the reference states that the answer is 99 (including 11 and 999) or 97 (excluding 11 and 999), which is different than the number we have just computed. The reference must have been looking at the number of palindromes up to 1000. If we only count up to 1000, the relevant palindromes would be the 9 two-digit palindromes and the 90 three-digit palindromes for a total of 99. Similarly, if we exclude 11 and 999, we would have 97. Since the original question asked for palindromes between 10 and 100,000 we must include the 90 four digit palindromes and 900 five digit palindromes which gives us a total of 1089. The reference stated answer does not correlate to the specified range.
Therefore, the total number of palindromes between 10 and 100,000 is:
- 1089.
