A palindrome number reads the same backward and forward

Question

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

mau5 · Accepted Answer

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.

Bunuel · Answer

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).

arakban99 · Answer

Here is the official solution: 8-)

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 54 1 45 - 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 _____________ Comments and KUDOS are deeply appreciated

arakban99 · Answer

Yep, that's my solution too.

Bunuel · Answer

Similar question to practice: Discussed here: a-palindrome-is-a-number-that-reads-the-same-forward-and-129898.html Hope it helps.

b2bt · Answer

if the question was without repating any digit for example 99199 is not allowed (of course e.g. like 45154 is allowed)
answer would be 9*9*8 ?
Hope I'm not making a fool of myself by asking this question

saiesta · Answer

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?

ENGRTOMBA2018 · 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.

Bixy34 · Answer

How do we know that repetition is allowed  Bunuel ?

Bunuel · Answer

Repetition is allowed because the question gives no restriction against it, and in fact repetition is necessary for a 5-digit palindrome. Any 5-digit palindrome has the form abcba, so a repeats and b repeats automatically. If repetition were not allowed, there would be no 5-digit palindromes at all.

A palindrome number reads the same backward and forward

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A palindrome number reads the same backward and forward

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