A palindrome is a number
that reads the same from right to left as it does from left to right. For
example, 363, 77, and 24642 are palindromes. How many four-digit palindromes
involves the use of permutations: how many different ways can you arrange
members of a set. Here the set referred to is the set of digits one through
nine and zero.
Plans to Solve/Investigate
As we determine the total
number of four-digit palindromes possible, we actually only need consider the
first two digits as digits three and four create the palindrome. The first
digit of the set of solutions to the problem can be one of nine different digits
(for you cannot begin a four digit with zero). The solution set for the second
digit can be from one of ten digits (for you can have zero as the second digit).
of the Problem
Beginning with one as the
first digit, there are ten possible palindromes.
(1111 works as eleven is
still eleven reversed). There are no other possible arrangements of the first
two digits while limiting the first digit to one. Beginning with two through
nine, there are also ten palindromes per first digit. Nine different first
digits * ten different palindromes per first digit = ninety different
The key to understanding
the solution to this problem lies in understanding you actually need only
consider the first two digits. For any two-digit number (not beginning with
zero) one can write those two digits in reverse order to create a four-digit
Another way to look at the
The first two digits can
be all numbers between and including 10 and 99. That is a total of ninety
numbers. As each of those two digits can represent a different four-digit
palindrome, you therefore have ninety palindromes.
Extensions of the Problem
A good extension to the
problem, would be to have students find all the possible license plates in a
state that have a sequence of three letters followed by three numbers. How many
license plates could start with MOM or POP?