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Question

A palindrome is a number that reads the same forwards and backwards, such as 12321. How many 5-digit palindromes are there?

Answer 1

The term "palindrome" may scare you (because it is new), but it is an easy concept.  This is a problem about combinations.

 

The key to solving this problem is to use the right method to break it down into easy problems.

 

First, how many digits may be used?  10   (0-9)

 

Then, let's append (oops! another unusual word) a digit to the end.  For each first digit, how many digits may be place on the right of it?   10      So, now we have 10*10 possibilities    [00, 01, 02, 03, 04, ..., 99]

 

And, how many third digits can be appended to each of the previous two-digit values?  10

    That makes 10*10*10 possibilities    [000, 001, 002, ... 999; oh, you can count]

 

Consider the word "palindrome" again.  We have selected all the possibilities for a 5-digit palindrome because it has only 3 unique digits.    [cbAbc and we have Abc already]

 

Now, you must decide whether a leading "0" is allowed (that is, should "01210" be considered?).  If you view this like an odometer or a combination lock or a PIN number or ...   then, yes, it should.

 

O.K., how many combinations was that??