Subject: palindromes.

Who is asking: Student
Level: Secondary

Positive integers such as 1287821 and 4554, in which the number is unchanged when the digits are reversed, are called palindromes. The number of five-digit integers larger than or equal to 10,000 which are not palindromes is... a. 10 000 b. 81 000 c. 89100 d. 90 000 e. 99 100

Thanks for taking the time to help me out.

It is easier to count the numer that are palindromes. If you start to construct a five digit palindrome then the first digit can be any digit from 1 to 9. That is, there are nine choices for the first digit, which must then also be the fifth digit.

For the second digit you can choose any digit from 0 to 9 so there are ten choices for the second digit. This digit must also be placed in the fourth position if the number is to be a palindrome.

By the same reasoning there are ten choices for the third digit and this completes the number. Thus there are 9 x 10 x 10 = 900 choices for the first three digits and hence 900 five digit palindromes greater than or equal to 10,000.

Since there are 9,000 five digit integers greater than or equal to 10,000 there are 9,000 - 900 = 8,100 five digit palindromes greater than or equal to 10,000.

In May 2020 we got a message from Maurizio that my solution is incorrect. Maurizio is correct. The paragraph directly above says "Since there are 9,000 five digit integers greater than or equal to 10,000..." but in fact there are 90,000 five digit integers greater than or equal to 10,000. Hence the answer to Jacky's question should be "there are 90,000 - 900 = 89,100 five digit palindromes greater than or equal to 10,000."

Thanks Maurizio.