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Manager  Status: Still Struggling
Joined: 03 Nov 2010
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GMAT Date: 10-15-2011
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A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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3
12 00:00

Difficulty:   45% (medium)

Question Stats: 62% (01:34) correct 38% (01:50) wrong based on 394 sessions

### HideShow timer Statistics A palindrome is a number that reads the same forward and backward, such as 242. How many even five-digit numbers are palindromes?

A. 40
B. 400
C. 500
D. 5,000
E. 100,000

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

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7
1
Hi Anamika,

The best strategy to attack this problem is to just Dive In.

What do we know? From the definition of "palindrome", we know that the five digits must be arranged in such a way that they form the pattern: ABCBA.

Because the question is asking for an EVEN number, the number in position A can only be 2, 4, 6, or 8. This gives us 4 unique options.
Both B and C can be any one of the 10 digits (0-9).

Now, in the first position, we can have 4 options and the second and third can have 10 each. The fourth and fifth positions can only have 1 option (that is whatever has already been determined in the first and second positions).

Multiplying this through to find the total number of combinations, we get 4*10*10*1*1 = 400.

Hope that helps! Good luck studying.

------------
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##### General Discussion
Intern  Joined: 02 Aug 2011
Posts: 5
Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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1
A palindrome is a number that reads the same forward and backward, such as 242. How many even five-digit numbers are palindromes?
40
400
500
5,000
100,000[/quote]

the answer is B ( 400) . here's the solution..
if the plindrome must be even, it must begin and end in 2,4 6 and 8 ( 4 ways) e.g 22122,44144.
The middle number in the five-digit could be 0-9 ( 10 ways), e.g 22522, 22822
The 2nd and fourth numbers could must also be same i.e 0-9( 10 ways) e.g 23432, 85658
we thus have, 4 *10*10 = 400.
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Joined: 24 Jul 2011
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GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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(B) it is. Here is a different solution:

Choose one number for the 1st and 5th digits from 2,4,6,or 8. This number must be even and cannot be 0 because if the first digit is 0, it would make the 5 digit number a 4 digit number. This selection can be done in 4C1 ways.

Choose one number for the second and fifth digits. This can be any one of the ten digits. 10C1 ways.

Choose one number for the middle digit (3rd digit). This can again be done in 10C1 or 10 ways.

Therefore total number of ways = 4C1 * 10C1 * 10C1 = 400
=> (B).
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Manager  Joined: 06 Feb 2011
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Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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1
Missed out the word even from the question, in a hurry. Got answer as 900.
Director  Joined: 01 Feb 2011
Posts: 629
Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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5th digit 2,4,6,8
4th digit 0 through 9
3rd digit 0 through 9

i.e 10*10*4 = 400 numbers

first and second digit is going to be same as 4th and 5th. so it would still be 400 numbers.

Director  Joined: 03 May 2007
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Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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krishnasty wrote:
A palindrome is a number that reads the same forward and backward, such as 242. How many even five-digit numbers are palindromes?
40
400
500
5,000
100,000

Agree with B.

The first and last digits can only be even integers and are the same in 4 ways = 4c1
The second and fourth digits are the same but ranges from 0 -9 in 10 ways = 10c1
The third digit could be any integer from 0 - 9 in 10 ways = 10c1

So the desired number of 5 digit palindromes = 4c1 x 10c1 x 10c1 = 400
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Re: A palindrome is a number that reads the same forward and backward,  [#permalink]

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1
Anamika2014 wrote:
A palindrome is a number that reads the same forward and backward, such as 242. How many even five-digit numbers are palindromes?

A. 40
B. 400
C. 500
D. 5,000
E. 10,000

Similar questions to practice:

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

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1
palindrome five digit sequence.
abcde.
a equals e , b equals d , c can be any single digit number 1 to 10.
so c can be filled 10ways.

even five digit numbers,so last digit e can take 0,2,4,6 and 8.
a can't take 0 as first digit , so a and e can fill in four ways.

b and d can fill in ten ways

So abc can fill in 4*10*10 . total 400ways .

option B is correct.
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Posts: 7764
Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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krishnasty wrote:
A palindrome is a number that reads the same forward and backward, such as 242. How many even five-digit numbers are palindromes?

A. 40
B. 400
C. 500
D. 5,000
E. 100,000

Lets work on the digits one by one -
1) UNITS digit
= tthe ten thousands digit = even number = 2,4,6,8--- 4 ways............) is not counted because 0 at ten thusands place will make it a 4-digit number..
2) TENS digit = same as thosands digit = any of 10 digits...
3) HUNDREDS digit = any of 10 digits ..

Total = 4*10*10 = 400
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Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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chetan2u wrote:
krishnasty wrote:
A palindrome is a number that reads the same forward and backward, such as 242. How many even five-digit numbers are palindromes?

A. 40
B. 400
C. 500
D. 5,000
E. 100,000

Lets work on the digits one by one -
1) UNITS digit
= tthe ten thousands digit = even number = 2,4,6,8--- 4 ways............) is not counted because 0 at ten thusands place will make it a 4-digit number..
2) TENS digit = same as thosands digit = any of 10 digits...
3) HUNDREDS digit = any of 10 digits ..

Total = 4*10*10 = 400

Why have you not considered 4 & 10 for the ten thousands digit and thousands digits respectively?
I was thinking in the lines of 4 x 10 x 10 x 10 x 4.
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Joined: 02 Aug 2009
Posts: 7764
Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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1
nishi999 wrote:
chetan2u wrote:
krishnasty wrote:
A palindrome is a number that reads the same forward and backward, such as 242. How many even five-digit numbers are palindromes?

A. 40
B. 400
C. 500
D. 5,000
E. 100,000

Lets work on the digits one by one -
1) UNITS digit
= tthe ten thousands digit = even number = 2,4,6,8--- 4 ways............) is not counted because 0 at ten thusands place will make it a 4-digit number..
2) TENS digit = same as thosands digit = any of 10 digits...
3) HUNDREDS digit = any of 10 digits ..

Total = 4*10*10 = 400

Why have you not considered 4 & 10 for the ten thousands digit and thousands digits respectively?
I was thinking in the lines of 4 x 10 x 10 x 10 x 4.

Hi Nishi,

the reason is that THOUSAND's digit has to be same as TEN's digit and 10,000's digit should be same as units digit, as the number is a palindrome....
so if you take ten's digit as 2, thousands automatically becomes 2.....
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Re: A palindrome is a number that reads the same forward and backward,  [#permalink]

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A palindrome is a number that reads the same forward and backward,  [#permalink]

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Must be careful to discount 0 as the first digit. Although 0 is even, if 0 is the first digit, the number becomes a 4 digit number!!

4x10x10=400.
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Joined: 16 Jul 2018
Posts: 43
Re: A palindrome is a number that reads the same forward and backward,  [#permalink]

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I was wondering if you could help me with this problem... My way of approach is that there are 3 different cases.
1) aaaaa --->all even and same = 4 ways (2222,4444 etc.)
2) aabaa---->first 2 and last 2 same = 144 ways eg. 22322
3) abcba ----> 324 ways eg. 21312
4+144+324=472
What is the flaw of my approach? I have wrongly calculated the combinations in each case or is it something else?
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GMAT 1: 740 Q50 V40 GMAT 2: 770 Q51 V42 Re: A palindrome is a number that reads the same forward and backward,  [#permalink]

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1
UNSTOPPABLE12, Happy to help This question can be solved in a matter of seconds by taking a systematic approach.

Let us say the five digit number is _ _ _ _ _
1. For a number to be a five digit number, the first digit cannot be zero (else it will be a four digit number)
2. When we choose the last two blanks, we will have no choice left for the first two blanks - as they will be identical to the last two - this is the definition of the palindrome.
3. The central number ( hundredths place) can be anything.
4. The ending digit can be an even number other than zero - keeping point 1 in mind.

Based on the above three criteria we have 1 1 10 10 4 ways to choose the _ _ _ _ _ respectively.

Hence a total of 1*1*10*10*4 ways or 400 ways - using the AND-rule.

Hope this all makes sense.

UNSTOPPABLE12 wrote:

I was wondering if you could help me with this problem... My way of approach is that there are 3 different cases.
1) aaaaa --->all even and same = 4 ways (2222,4444 etc.)
2) aabaa---->first 2 and last 2 same = 144 ways eg. 22322
3) abcba ----> 324 ways eg. 21312
4+144+324=472
What is the flaw of my approach? I have wrongly calculated the combinations in each case or is it something else?

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“Do. Or do not. There is no try.” - Yoda (The Empire Strikes Back) Re: A palindrome is a number that reads the same forward and backward,   [#permalink] 09 Jan 2019, 11:57
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