Problem Solving

A palindrome is a number that reads the same forward and backward. For example. 2442 and 111 are palindromes. If 5-digit palindromes are formed using one or more of the digits, 1, 2, 3, how many such palindromes are possible?

A) 12
B) 15
C) 18
D) 24
E) 27

Take the task of building palindromes and break it into stages.

Stage 1: Select the ten-thousands digit
We can choose 1, 2, or 3
So, we can complete stage 1 in 3 ways

Stage 2: Select the thousands digit
We can choose 1, 2, or 3
So, we can complete stage 2 in 3 ways

Stage 3: Select the hundreds digit
We can choose 1, 2, or 3
So, we can complete stage 3 in 3 ways

IMPORTANT: At this point, the remaining digits are already locked in.

Stage 4: Select the tens digit
This digit must be the SAME as the thousands digit (which we already chose in stage 2)
So, we can complete this stage in 1 way.

Stage 5: Select the units digit
This digit must be the SAME as the ten-thousands digit (which we already chose in stage 1)
So, we can complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus build a 5-digit palindrome) in (3)(3)(3)(1)(1) ways ([spoiler]= 27 ways[/spoiler])

Answer: E
--------------------------

Note: the FCP can be used to solve the majority of counting questions on the GMAT. For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html
- https://www.beatthegmat.com/mouse-pellets-t274303.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/ps-counting-t273659.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/please-solve ... 71499.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/laniera-s-co ... 15764.html

Cheers,
Brent

Hi prata,

There's a full discussion of this question here:

https://www.beatthegmat.com/palindrome-5 ... 85219.html

GMAT assassins aren't born, they're made,
Rich