2009 AMC 10B Problem 11

Question

How many

7

-digit palindromes (numbers that read the same backward as forward) can be formed using the digits

2,

2,

3,

3,

5,

5,

5?

6

12

24

36

48

In a

7

-digit palindrome the middle digit appears an odd number of times

The first three digits determine the last three; count their orderings

A

7

-digit palindrome has the form with the middle digit used once and the outer three digits each used twice. Only

5

appears an odd number of times, so

5

must be the middle digit.

The remaining digits

2,3,5

fill the first three positions in some order and mirror to the last three. There are

3!=6

such orderings.

Thus, the correct answer is A.

Problem 11 in other years