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# A palindrome is a number that reads the same forward and bac

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Joined: 29 May 2012
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A palindrome is a number that reads the same forward and bac  [#permalink]

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Updated on: 31 Oct 2013, 04:47
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104
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Difficulty:

45% (medium)

Question Stats:

63% (01:20) correct 37% (01:59) wrong based on 933 sessions

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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

Originally posted by accincognito on 07 Oct 2013, 00:31.
Last edited by accincognito on 31 Oct 2013, 04:47, edited 4 times in total.
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Joined: 02 Sep 2009
Posts: 64224
Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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07 Oct 2013, 00:37
42
42
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.

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Hope this helps.
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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27 Jul 2014, 18:04
12
1
2
There are 3 options for the 1st digit, so this implies that there is 1 option for the 5th digit, because it must be the same as the 1st digit.
There are 3 options for the 2nd digit, so this implies that there is 1 option for the 4th digit, because it must be the same as the 2nd digit.
There are 3 options for the 3rd digit (because it is the middle).

3*3*3*1*1 = 27 options
##### General Discussion
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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07 Oct 2013, 20:46
6
Started with 1:

12321
13231
12221
13331
11111
11311
11211
12121
13131

9 possibilities for 1, 9 for 2, and 9 for 3 =
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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30 May 2016, 21:10
4
1
Since this is a 5 digit palindrome, the middle number would act like a mirror.

Case 1: Middle number = 1
_ _ 1 _ _
Number of options for 1st place = 3
Number of options for 2nd place = 3
Number of options for 4th and 5th place = 1
Total ways = 3*3 = 9

Case 2: Middle number = 2
_ _ 2 _ _
Similar to the above reasoning,
Total ways = 3*3 = 9

Case 3: Middle number = 3
_ _ 3 _ _
Similar to the above reasoning,
Total ways = 3*3 = 9

Total number of ways = 9 + 9 + 9 = 27

Correct Option: E
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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08 Jun 2014, 18:54
3
1
I did the counting method. Was able to do it under 2 minutes.
Since it is a palindrome, I only counted the first 3 digits since the last 2 would mirror the initial 2 digits.

- All possible permutations of XYZ (All digits different) --> 6
- All possible permutations of XXZ (1st & 2nd Digit are the same) --> 6
- All possible permutations of XZZ (1st & 3rd Digit are the same) --> 6
- All possible permutations of XYX (1st & 3rd Digit are the same) --> 6
- All possible permutations of XXX (All digits same) --> 3

n=6+6+6+6+3=27 --E
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A palindrome is a number that reads the same forward and bac  [#permalink]

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10 Oct 2013, 00:50
1
6
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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08 Jun 2014, 01:08
1
Hi Bunuel,

Can you please explain the logic behind the same? There are many kinds of palindromes that can be formed using two or three numbers (e.g. 12221, 12121, 12321, 32123) I am unable to understand how your approach caters to all of them?

Thank you

Regards,
Neha

Bunuel wrote:
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.

Similar questions to practice:

Hope this helps.
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Posts: 64224
Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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08 Jun 2014, 02:46
1
nehabansal123 wrote:
Hi Bunuel,

Can you please explain the logic behind the same? There are many kinds of palindromes that can be formed using two or three numbers (e.g. 12221, 12121, 12321, 32123) I am unable to understand how your approach caters to all of them?

Thank you

Regards,
Neha

Bunuel wrote:
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.

Similar questions to practice:

Hope this helps.

In a five-digit palindrome XYZYX we are concerned about X, Y and Z, so about the first three digits.

You get 12221 if you choose 1, for X, 2 for Y and 2 for Z, the same for other options.
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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08 Jun 2014, 03:50
1
Voila! Makes complete sense, thank you so much for responding!

Bunuel wrote:
Hi Bunuel,

Can you please explain the logic behind the same? There are many kinds of palindromes that can be formed using two or three numbers (e.g. 12221, 12121, 12321, 32123) I am unable to understand how your approach caters to all of them?

Thank you

Regards,
Neha

Bunuel wrote:
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.

Similar questions to practice:

Hope this helps.

In a five-digit palindrome XYZYX we are concerned about X, Y and Z, so about the first three digits.

You get 12221 if you choose 1, for X, 2 for Y and 2 for Z, the same for other options.[/quote]
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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30 Aug 2015, 06:52
1
I can choose out of (1,2 and 3) for the first value.
I can choose out of (1,2 and 3) for the second value and
I can also choose out of (1,2 and 3) for the third value.

However: To get a "palindrome" number the two proceeding terms have to read the same as the first two. As I have already chosen values for the first and the second term I logically have only one choice for the next two, which is the one that I already chose.

Therefore it is: 3 * 3 * 3 * 1 * 1 = 27 possible combinations
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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21 Nov 2016, 23:34
1
Top Contributor
accincognito wrote:
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

Let us take one case

1---1

The inside 3 digits can be formed with one digit occurring thrice i.e, 111, 222, 333 in 3 ways
And with one digit occurring once i.e, 121, 131, 212, 232, 313, 323 in 6 ways

So there are 9 ways with 1 at the ends

Totally 27 ways considering 1, 2 and 3
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A palindrome is a number that reads the same forward and bac  [#permalink]

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15 Jun 2017, 07:47
_ _ fist two numbers

1 1
1 2
1 3
2 1
2 2
2 3
3 1
3 2
3 3

9 options

_ middle number

we can change this number, thus 3 options

_ _ _ last three numbers

they correspond to the first 3 numbers

1 option

9*3*1 = 27

therefore 5*3*1=27
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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15 Mar 2019, 17:50
accincognito wrote:
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

We have 3 options for the first digit, 3 options for the second, 3 options for the third, 1 option for the fourth (since it has to be the same as the second), and 1 option for the fifth (since it has to be the same as the first). Thus, there are 3 x 3 x 3 = 27 possible palindromes.

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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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03 Apr 2020, 01:55
Take a very easy approach here...

5 digit palindrome of ABCDE being 5 digits
and by definition since should be the same backwards A=E, B=D
therefore, we get ABCBA as 5 digits

options being 1,2,3

Therefore A can be =1,2,3 ( 3 options)
B can also be = 1,2,3 (3 Options)
C can be 1,2,3 (3 options)

=> 3 x 3 x3 = Total number of outcomes possible. = 27 Answer E

Had we not been allowed to use a digit more than twice, then our outcome could have been 3 x 2 x 1 = 6 (Maybe) please comment your views
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Re: A palindrome is a number that reads the same forward and bac  [#permalink]

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31 May 2020, 11:26
Hello

I get the idea while counting the possibilities....
However if we want to coin the concept or formula.. could we say that it is like distributing different objects to different bins? ( the term coined to a certain type of combination problems)
so we say
3 digits put in 3 bins... we get same result
Re: A palindrome is a number that reads the same forward and bac   [#permalink] 31 May 2020, 11:26