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Source: — Problem Solving |
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by GMATGuruNY » Wed Jul 09, 2014 2:54 am
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
To read the same forward and backward, the 5-digit integer must look as follows:
ABCBA.
The ten-thousands digit and the units digit must be THE SAME.
The thousands digit and the tens digit must also be THE SAME.

Number of options for the ten-thousands digit = 3. (1, 2, or 3)
Number of options for the units digit = 1. (Must be the same as the ten-thousands digit)
Number of options for the thousands digit = 3. (1, 2, or 3)
Number of options for the tens digit = 1. (Must be the same as the thousands digit)
Number of options for the hundreds digit = 3. (1, 2, or 3)
To combine these options, we multiply:
3*3*3*1*1 = 27.

The correct answer is E.
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by Brent@GMATPrepNow » Wed Jul 09, 2014 5:43 am
Here's another (slightly trickier) palindrome question to practice with: https://www.beatthegmat.com/palindrome-t263705.html

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by Brent@GMATPrepNow » Wed Jul 16, 2014 6:51 pm
Here's a palindrome question I created for BTG a longggg time ago (for their Math Challenge Question contest):
A palindrome is a word that is read the same backwards as forwards. For example, the words "BADAB," "IAGAI," and "HHHHH" are all palindromes.

How many 5-letter palindromes can be created using the letters A, B, C, D, E, F, G, H, I and J?
For a full solution, watch the following YouTube video: https://youtu.be/qfiPnXIBx7g

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by Brent@GMATPrepNow » Wed Jul 16, 2014 6:56 pm
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
Brent Hanneson - Creator of GMATPrepNow.com
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