A palindrome number reads the same backward and forward : GMAT Problem Solving (PS)
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# A palindrome number reads the same backward and forward

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06 Sep 2013, 04:54
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A palindrome is a number which reads the same when read forward as it it does when read backward. So, how many 5 digit palindromes are there?

(A) 720
(B) 800
(C) 890
(D) 900
(E) 950
[Reveal] Spoiler: OA

Last edited by arakban99 on 06 Sep 2013, 05:10, edited 1 time in total.
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06 Sep 2013, 05:04
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arakban99 wrote:
A palindrome is a number which reads the same when read forward as it it does when read backward. So, how many 5 digit palindromes are there?

(A)720
(B)800
(C)890
(D)900
(E)950

A 5 digit palindrome would look like this : $$abcba.$$
Thus, the first digit can be filled in 9 ways(1-9), the second digit can be filled in 10 ways (0-9) and the 3rd digit can again be filled in 10 ways. The last 2 digits would just mirror the already selected digit. Thus, the no of ways : 9*10*10*1*1 = 900.
D.
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06 Sep 2013, 05:04
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Here is the official solution:

there five digit places

__ __ __ __ __

The last two depend on the first two (because the last to is the flipped version of the first two digits i.e 54145 - 45 is the flipped version of 54 so the number of ways to select the last two is 1*1

So this the total of ways to select (combination) the digits: 9 (digits 1-9, remember 0 at the start will make the number a four digit number) * 10 (0-9) * 10 (0-9) * 1 *1

= 900.

Hope it helped
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Last edited by arakban99 on 06 Sep 2013, 05:07, edited 1 time in total.
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06 Sep 2013, 05:06
mau5 wrote:
arakban99 wrote:
A palindrome is a number which reads the same when read forward as it it does when read backward. So, how many 5 digit palindromes are there?

(A)720
(B)800
(C)890
(D)900
(E)950

A 5 digit palindrome would look like this : $$abcba.$$
Thus, the first digit can be filled in 9 ways(1-9), the second digit can be filled in 10 ways (0-9) and the 3rd digit can again be filled in 10 ways. The last 2 digits would just mirror the already selected digit. Thus, the no of ways : 9*10*10*1*1 = 900.
D.

Yep, that's my solution too.
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06 Sep 2013, 06:33
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Expert's post
arakban99 wrote:
A palindrome is a number which reads the same when read forward as it it does when read backward. So, how many 5 digit palindromes are there?

(A)720
(B)800
(C)890
(D)900
(E)950

_____________
Comments and KUDOS are deeply appreciated

Similar question to practice:
Quote:
A palindrome is a number that reads the same forward and backward, such as 121. How many odd, 4-digit numbers are palindromes?

A. 40
B. 45
C. 50
D. 90
E. 2500

Hope it helps.
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06 Sep 2013, 07:29
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if the question was without repating any digit for example 99199 is not allowed (of course e.g. like 45154 is allowed)
Hope I'm not making a fool of myself by asking this question
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07 Sep 2013, 04:21
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b2bt wrote:
if the question was without repating any digit for example 99199 is not allowed (of course e.g. like 45154 is allowed)
Hope I'm not making a fool of myself by asking this question

Yes, if repetition were not allowed, then the answer would be 9*9*8 --> abcba:

9 options for a (from 1 to 9 inclusive);
9 options for b (from 0 to 9 inclusive minus the digit we used for a);
8 options for c (from 0 to 9 inclusive minus two digit we used for a and b).
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13 Nov 2014, 10:35
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30 Nov 2015, 22:05
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16 Feb 2016, 10:22
mau5 wrote:
arakban99 wrote:
A palindrome is a number which reads the same when read forward as it it does when read backward. So, how many 5 digit palindromes are there?

(A)720
(B)800
(C)890
(D)900
(E)950

A 5 digit palindrome would look like this : $$abcba.$$
Thus, the first digit can be filled in 9 ways(1-9), the second digit can be filled in 10 ways (0-9) and the 3rd digit can again be filled in 10 ways. The last 2 digits would just mirror the already selected digit. Thus, the no of ways : 9*10*10*1*1 = 900.
D.

I understand that a 5 digit palindrome would look like $$abcba$$. However, a 5 digit palindrome could also look like $$aaaaa$$ or $$ababa$$. Why are we not adding the combinations of these two palindromes to our answer?
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16 Feb 2016, 10:26
saiesta wrote:
mau5 wrote:
arakban99 wrote:
A palindrome is a number which reads the same when read forward as it it does when read backward. So, how many 5 digit palindromes are there?

(A)720
(B)800
(C)890
(D)900
(E)950

A 5 digit palindrome would look like this : $$abcba.$$
Thus, the first digit can be filled in 9 ways(1-9), the second digit can be filled in 10 ways (0-9) and the 3rd digit can again be filled in 10 ways. The last 2 digits would just mirror the already selected digit. Thus, the no of ways : 9*10*10*1*1 = 900.
D.

I understand that a 5 digit palindrome would look like $$abcba$$. However, a 5 digit palindrome could also look like $$aaaaa$$ or $$ababa$$. Why are we not adding the combinations of these two palindromes to our answer?

The cases you are mentioning are already covered in the solution above. Lets say you have the form of abcba, then for the first digit you have 9 ways, for second digit ('b') you have 10 digits and finally for c again, you will have 10 ways giving you a total of 9*10*10*1*1 = 900 ways.

aaaaa and ababa are already covered in the above scenarios for example lets say, a = 5, then for b we can again have 5 as we have all 10 digits allowed. Repeat the same for 'c' and you will get your aaaaa combination. Same logic applies to ababa as well.

Hope this helps.
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Re: A palindrome number reads the same backward and forward   [#permalink] 16 Feb 2016, 10:26
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