Palindrome

Probability Level 2

A palindrome is a number (with no leading zeros) that remains the same when written backwards. For a 3-digit palindrome,

  • the first digit has 9 options (1 to 9),
  • the second digit has 10 options (0 to 9), and
  • the third digit has 1 option (must be the same as the first digit).

By the rule of product, the number of all 3-digit palindromes is 9 × 10 × 1. 9\times 10 \times 1.

True or False?

By the same token, the number of all 5-digit palindromes is 9 × ( 9 × 10 × 1 ) × 1 , 9\times (9\times 10\times 1)\times 1, because we simply need to add the same digit (1 to 9) to both the front and back of each of the 3-digit palindromes obtained above.

True False

Christopher Boo by Christopher Boo , 24, Singapore

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1 solution

Marta Reece
Jul 7, 2017

Five digits palindromes can have a zero as a second digit. Those palindromes cannot be generated by adding to three digit palindromes.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

The wording is not ideal. You are saying "If we add the same digit to the front and back of the palindrome, there are 9 × 9 × 10 × 1 × 1 9\times9\times10\times1\times1 five digit palindromes." If we do that, and only that, there will indeed be that many palindromes. So the question is ambiguous. Are we restricted to that procedure or not?

I would phrase it somewhat like: "We can generate five digit palindromes by adding the same digit to the front and back of a three digit palindrome. Does this mean that there are 9 × 9 × 10 × 1 × 1 9\times9\times10\times1\times1 five digit palindromes?"

Marta Reece - 3 years, 11 months ago

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