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A palindrome is a number that reads the same forward and bac 04 Sep 2020, 07:40
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Given

    • A palindrome is a number that reads the same forward and backward.
    • 5-digit palindromes are formed using one or more of the digits, 1, 2, 3.

To Find

    • The number of palindromes.


Approach and Working Out

    • The general from can be ABCBA.

    • So here just need to assign A, B, C the last two digits will be assigned automatically.
      o Total number of ways = 3 × 3 × 3 = 27

Correct Answer: Option E
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A palindrome is a number that reads the same forward and bac 16 Oct 2021, 02:20
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HouseStark
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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

1st 2nd 3rd 4th 5th
___ ___ ___ ___ ____

The 1st place can be filled by (1,2,3)= \(3 \)ways

also, The 2nd and 3rd place can be filled by (1,2,3)=\(3\) ways(for both 2nd and 3rd place)

the 4th place digit has to be same with the digit in 2nd place and 5th place digit has to be same with the digit in 1st place to make a palindrome =1 ways (for both 4th and 5th place)

total =\(3*3*3*1*1=27 \)

Answer E
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Hey! I understood your explanation, i just have one doubt.
In the slot method, the digits could be shuffled right, unless we divide the whole thing by 5!

So, here, if we are not dividing it by 5!, then how are we getting rid of the shuffle possibility?
Experts, pls help

Bunuel VeritasKarishma ScottTargetTestPrep
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A palindrome is a number that reads the same forward and bac 17 Oct 2021, 23:02
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Chitra657
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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

1st 2nd 3rd 4th 5th
___ ___ ___ ___ ____

The 1st place can be filled by (1,2,3)= \(3 \)ways

also, The 2nd and 3rd place can be filled by (1,2,3)=\(3\) ways(for both 2nd and 3rd place)

the 4th place digit has to be same with the digit in 2nd place and 5th place digit has to be same with the digit in 1st place to make a palindrome =1 ways (for both 4th and 5th place)

total =\(3*3*3*1*1=27 \)

Answer E

Hey! I understood your explanation, i just have one doubt.
In the slot method, the digits could be shuffled right, unless we divide the whole thing by 5!

So, here, if we are not dividing it by 5!, then how are we getting rid of the shuffle possibility?
Experts, pls help

Bunuel VeritasKarishma ScottTargetTestPrep
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Note that we are not using 3*3*3*3*3 which will lead to all possibilities of making a 5 digit number with 3 digits.
It will include 12321 (a palindrome) as well as 12231 (not a palindrome) i.e. all variations of 11223.

We are using 3*3*3*1*1 to give the spots.
The first spot can be filled in 3 ways. Say we choose 1 to fill the first spot. Then the last spot must have 1 too hence can be filled in only 1 way.
1 __ __ __ 1

Say we choose 2 for the second spot. Then the second last spot must have 2 too hence can be filled in only 1 way.
12 __ 21

Then say we choose 3 for the third spot. We get 12321.

This will be a palindrome in each case because we have ensured that first spot = fifth spot and second spot = fourth spot.
That is why 3*3*3*1*1 need not be divided by anything.
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A palindrome is a number that reads the same forward and bac 29 May 2022, 03:06
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Bunuel
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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

XYZYX

X can be 1, 2, or 3, thus 3 options.
Y can be 1, 2, or 3, thus 3 options.
Z can be 1, 2, or 3, thus 3 options.

Total 3^3=27.

Answer: E.

Similar questions to practice:
https://gmatclub.com/forum/a-palindrome- ... 29898.html
https://gmatclub.com/forum/a-palindrome- ... 59265.html

Hope this helps.
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Hi experts Bunuel IanStewart

This question took me about three minutes in my practice exam, but I still got it incorrectly.
I was not familiar with palindrome, and I did not see that the number of combinations could be obtained by 3*3*3. Instead, I tried to count the possibilities manually. (The answer choices range between 12 to 27, which is not too many to count. It is not the most efficient way, but I had no other ways then.) I failed because I only thought of three types of arrangements, AAAAA, ABCBA and ABBBA, ignoring other two, AABAA and ABABA.

I checked the solutions in this thread (they are smart) and practiced other two similar palindrome questions.
I hope to confirm two issues:

1. When a palindrome has an even number of digits, we only need to care about the first half. When the palindrome has an odd number of digits, we care about the first half and the middle one.

For example, for the 4-digit palindrome XYZW, we only chose numbers for X and Y, since Z and W just copy the preceding digits respectively.
For the 5-digit palindrome ABCDE, we only chose numbers for A,B and C, letting D and E copy A and B.

2. These palindrome questions may vary in constraints, such as "the number must be odd," "no digits can be repeated," "the digits could only be 1, 2 and 3." We just tackle each questions differently.


Thank you. :)
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A palindrome is a number that reads the same forward and bac 29 May 2022, 05:04
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Bunuel
paskorntt
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

XYZYX

X can be 1, 2, or 3, thus 3 options.
Y can be 1, 2, or 3, thus 3 options.
Z can be 1, 2, or 3, thus 3 options.

Total 3^3=27.

Answer: E.

Similar questions to practice:
https://gmatclub.com/forum/a-palindrome- ... 29898.html
https://gmatclub.com/forum/a-palindrome- ... 59265.html

Hope this helps.

Hi experts Bunuel IanStewart

This question took me about three minutes in my practice exam, but I still got it incorrectly.
I was not familiar with palindrome, and I did not see that the number of combinations could be obtained by 3*3*3. Instead, I tried to count the possibilities manually. (The answer choices range between 12 to 27, which is not too many to count. It is not the most efficient way, but I had no other ways then.) I failed because I only thought of three types of arrangements, AAAAA, ABCBA and ABBBA, ignoring other two, AABAA and ABABA.

I checked the solutions in this thread (they are smart) and practiced other two similar palindrome questions.
I hope to confirm two issues:

1. When a palindrome has an even number of digits, we only need to care about the first half. When the palindrome has an odd number of digits, we care about the first half and the middle one.

For example, for the 4-digit palindrome XYZW, we only chose numbers for X and Y, since Z and W just copy the preceding digits respectively.
For the 5-digit palindrome ABCDE, we only chose numbers for A,B and C, letting D and E copy A and B.

2. These palindrome questions may vary in constraints, such as "the number must be odd," "no digits can be repeated," "the digits could only be 1, 2 and 3." We just tackle each questions differently.


Thank you. :)
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You are correct about both points!!
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GraceSCKao

I was not familiar with palindrome, and I did not see that the number of combinations could be obtained by 3*3*3. Instead, I tried to count the possibilities manually. (The answer choices range between 12 to 27, which is not too many to count. It is not the most efficient way, but I had no other ways then.) I failed because I only thought of three types of arrangements, AAAAA, ABCBA and ABBBA, ignoring other two, AABAA and ABABA.

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I wouldn't suggest thinking of palindrome counting as a 'question type' (and thus learning some method for this precise situation). This might be the only official question that will ever test palindromic numbers. It's better to think of this question as just a standard counting problem where earlier choices constrain later choices. When we pick our first digit here, because the number is a palindrome, that uniquely determines the final digit, and when we pick the second digit that determines the fourth digit. That's a situation that could be tested in any number of ways -- for example, you could be asked how many six digit numbers there are where all of the odd-positioned digits are the same as each other, and all of the even-positioned digits are the same as each other, numbers like 373737. That's not a palindrome question, but it's testing the same idea that this palindrome question tests.

All of that said, the answer to both of your questions is 'yes'. :)
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A palindrome is a number that reads the same forward and bac 29 May 2022, 10:50
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I wouldn't suggest thinking of palindrome counting as a 'question type' (and thus learning some method for this precise situation). This might be the only official question that will ever test palindromic numbers. It's better to think of this question as just a standard counting problem where earlier choices constrain later choices. When we pick our first digit here, because the number is a palindrome, that uniquely determines the final digit, and when we pick the second digit that determines the fourth digit. That's a situation that could be tested in any number of ways -- for example, you could be asked how many six digit numbers there are where all of the odd-positioned digits are the same as each other, and all of the even-positioned digits are the same as each other, numbers like 373737. That's not a palindrome question, but it's testing the same idea that this palindrome question tests.

All of that said, the answer to both of your questions is 'yes'. :)
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IanStewart
Thank you for your explanations! :)

Your final question is really interesting.
Is the answer 9*9=81 if the digits in the odd and even positions could not be repeated?
If the digits could be repeated, the answer would be 9*10=90.
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A palindrome is a number that reads the same forward and bac 29 May 2022, 12:51
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GraceSCKao

Your final question is really interesting.
Is the answer 9*9=81 if the digits in the odd and even positions could not be repeated?
If the digits could be repeated, the answer would be 9*10=90.
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Yes, if I had taken more care to word the question well, it would have specified whether the odd and even placed digits could be equal, but both of your answers are correct.
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A palindrome is a number that reads the same forward and bac 02 Jul 2025, 05:52
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I understand the correction, but in the statement it says "If 5-digit palindromes are formed using one or more of the digits".
So for me, each of the integers 1, 2 or 3 must be part of the palyndrome at least once, which makes versions 12221 or 22122 where the 3 doesn't appear impossible, for example - no one agrees ?
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A palindrome is a number that reads the same forward and bac 02 Jul 2025, 07:58
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quengnst
I understand the correction, but in the statement it says "If 5-digit palindromes are formed using one or more of the digits".
So for me, each of the integers 1, 2 or 3 must be part of the palyndrome at least once, which makes versions 12221 or 22122 where the 3 doesn't appear impossible, for example - no one agrees ?
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The phrase "using one or more of the digits" does not mean each digit must appear. It just means digits must be chosen from the set {1, 2, 3}. So palindromes like 12221 or 11111 are valid.
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A palindrome is a number that reads the same forward and bac 20 Aug 2025, 14:31
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Consider this a question in which instead of 5 places you just have to fill 3 out of 5 position because for a number to be palindrome

-> first digit == last digit
-> second digit == last second digit


So now we have to only fill 3 places with integers 1, 2, 3.

_ , _ , _ , _ , _

a , b , c , b , a

For a, b, c we can fill with any of 1, 2, 3. Three possibilities for every digit.

Probability(Palindrome) = 3*3*3 = 27
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A palindrome is a number that reads the same forward and bac 31 Oct 2025, 23:25
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a question that took me a while to finish in gmat bc i didnt spot the pattern like most did but still got right <4mins

my initial solution: lay out all scenarios

scenario 1: same (x x x x x)
11111
22222
33333

scenario 2: one change, two numbers (x x y x x)
11211
11311
22122
22322
33133
33233

scenario 3: two changes, two numbers (x y x y x)
12121
13131
21212
23232
31313
32323

scenario 4: three changes, two numbers (x y y y x)
12221
13331
21112
23332
31113
32223

scenario 5: three changes, three numbers (x y z y x)
13231
12321
21312
23132
31213
32123

adding all five scenarios = 3 + 4*6 = (E)27

smarter solution: finding the patterns

we can find the pattern after writing down a few examples:
12321
23332
32123
33333

it's clear the only constraints are:
(1) 1st and 5th digits must be the same
(2) 2nd and 4th digits must be the same.
(3) only three options available: 1, 2, 3
(4) five digits in total

so we can map out the variables as pattern: x y z y x where x can = y and = z or not
in other words, we have 3 groups of variables each with 3 options to pick from -> 3*3*3 = 3^3 = (E)27
accincognito
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
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A palindrome is a number that reads the same forward and bac 08 Nov 2025, 12:06
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For the 5 digits, we can fill as 3 * 3 * 3 * 1 * 1 (last two digits are dependent on first two digits) = 27
accincognito
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
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