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

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Question Stats: 59% (01:12) correct 41% (01:23) wrong based on 874 sessions

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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

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

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16
16
enigma123 wrote:
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

@enigma123: Here is my explanation:

A palindrome is a number that reads the same forward and backward. Examples of four digit palindromes are 1221, 4334, 2222 etc
You basically get to choose the first two digits and you repeat them in opposite order. Say, you choose 45 as your first two digits. The next two digits are 54 and the number is 4554.
Also, you need only odd palindromes. This means that you need an odd digit at the end i.e. 1/3/5/7/9. This means that you need to start the number with an odd digit. Only then will it end with an odd digit.

In how many ways can you pick two digits such that the first one is an odd digit?
The first digit can be selected in 5 ways.(1/3/5/7/9)
The second digit can be selected in 10 ways.(0/1/2/3...8/9)
Total = 5*10 = 50 ways
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A palindrome is a number that reads the same forward and  [#permalink]

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

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enigma123 wrote:
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

First recognize you only need to consider the first two digits (because the second two are just the first two flipped)

There are 90 possibilities for the first two digits of a 4 digit number, 10-99 inclusive. Everything starting with a 1,3,5,7,9 will be odd, which is 5/9ths of the combinations.

5/9*90 = 50 C
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Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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The question demands that in Palindrome, the first digit must be odd that is why the last digit becomes odd.
For example, 1221 is a palindrome that reads same backward and forward and it is odd.
So the easy way of solving it is;
Take first odd digit, there are five odd digit and five different ways of selecting it
The second digit can be both odd or even no restriction, therefore there are 10 different ways of selecting it

The first digit can be selected in 5 ways.(1/3/5/7/9)
The second digit can be selected in 10 ways.(0/1/2/3...8/9)
Total = 5*10 = 50 ways

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

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pratikbais wrote:
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

We want to determine how many odd 4-digit numbers like XYYX are there. Notice here that in a palindrome X and Y can be the same, for example 1111 or 3333 are also palindromes.

Since, the number must be odd then the first and the last digits (X's) can take 5 values (1, 3, 5, 7, or 9) and the two middle digits (Y's) can take 10 values (0, 1, ..., 9), so total such numbers possible is 5*10=50.

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

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enigma123 wrote:
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

Similar questions to practice:

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

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HI Buneul,

A quick question with this problem. We can also make a palindrom out of event number also right.. such as 8228. Why does it have to be odd numbers.

Sorry a very old question.

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

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shriramvelamuri wrote:
HI Buneul,

A quick question with this problem. We can also make a palindrom out of event number also right.. such as 8228. Why does it have to be odd numbers.

Sorry a very old question.

pratikbais wrote:
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

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

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Bunuel wrote:
pratikbais wrote:
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

We want to determine how many odd 4-digit numbers like XYYX are there. Notice here that in a palindrome X and Y can be the same, for example 1111 or 3333 are also palindromes.

Since, the number must be odd then the first and the last digits (X's) can take 5 values (1, 3, 5, 7, or 9) and the two middle digits (Y's) can take 10 values (0, 1, ..., 9), so total such numbers possible is 5*10=50.

Hope it's clear.

Hi Bunuel,

what about negative numbers? Shouldn't we multiply the result by 2 ?
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Re: A palindrome is a number that reads the same forward and backward, suc  [#permalink]

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3
4 digits then the number must be in the format ABBA. A must be odd, so we have 5 possibilities for A. B can be whatever number, so it has 10 possibilities.
now, combine everything:
A = 5
B = 10
second B cannot be any of the 10 digits, it must be exactly as the first B, so it must be 1 combination
A = must be the same as first A, so there is only 1 combination
5*10*1*1 = 50
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Re: A palindrome is a number that reads the same forward and  [#permalink]

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Bunuel

Why is it not 2500? 5*10*10*5?
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Re: A palindrome is a number that reads the same forward and  [#permalink]

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4
1
Hi dubyap,

Since the 4th digit has to match the 1st digit and the 3rd digit has to match the 2nd digit, you don't have as many options as you might think.

Since the palindrome has to be ODD, both the 1st and 4th digits have to be ODD and the SAME...

The 1st digit could be 1, 3, 5, 7 or 9
The 4th digit must MATCH the first digit, so once you choose the 1st digit, there is ONLY ONE possible number for the 4th digit.

The 2nd digit could be any of the 10 options (0 - 9, inclusive).
The 3rd digit must MATCH the 2nd digit, so once you choose the 2nd digit, there is ONLY ONE possible number for the 3rd digit.

Thus, there are...

(5)(10)(1)(1) = 50 options

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

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The number is in the format ABBA, but note that B can equal A.

Choices for A = 1, 3, 5, 7, 9 = 5 choices
Choices for B = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 = 10 choices

5x10=50. Note that the third and fourth numbers are dictated by the second and first numbers respectively.
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Re: A palindrome is a number that reads the same forward and  [#permalink]

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enigma123 wrote:
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

The most convenient way to do this problem that I found was to consider the number of 4 digit numbers that begin and end with an odd number- not literally all of them but all four numbers must begin and end with 1,3,5,7,9 so if there were a matrix there would be five columns- how rows are there though, or rather possibilities under each column

There are five odd numbers that our four digit number can begin and end with hence 5 columns [1, 3, 5,7, 9]

Now,

1--1 the two numbers must match so actually from [0-9] there are ten possibilities

If there are 10 possibilities for each odd and there 5 odd numbers then there are 50 possibilities

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

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enigma123 wrote:
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

1001 1011 1021 1031 ... 1991
3003 3013 3023 3033 ... 3993

Bunuel, In between we can put 00-99, so there is 100 number.

Why the answer is not 500? 100(5)?
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Re: A palindrome is a number that reads the same forward and  [#permalink]

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hazelnut wrote:
enigma123 wrote:
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

1001 1011 1021 1031 ... 1991
3003 3013 3023 3033 ... 3993

Bunuel, In between we can put 00-99, so there is 100 number.

Why the answer is not 500? 100(5)?

Highlighted numbers are not palindromes. 1011 backwards is 1101, which is NOT the same number as 1011.
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Re: A palindrome is a number that reads the same forward and  [#permalink]

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Bunuel wrote:
pratikbais wrote:
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

We want to determine how many odd 4-digit numbers like XYYX are there. Notice here that in a palindrome X and Y can be the same, for example 1111 or 3333 are also palindromes.

Since, the number must be odd then the first and the last digits (X's) can take 5 values (1, 3, 5, 7, or 9) and the two middle digits (Y's) can take 10 values (0, 1, ..., 9), so total such numbers possible is 5*10=50.

Hope it's clear.

Hi Bunuel..

Can u tell me why we r doing 5*10 ways..

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

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zanaik89 wrote:
Bunuel wrote:
pratikbais wrote:
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

We want to determine how many odd 4-digit numbers like XYYX are there. Notice here that in a palindrome X and Y can be the same, for example 1111 or 3333 are also palindromes.

Since, the number must be odd then the first and the last digits (X's) can take 5 values (1, 3, 5, 7, or 9) and the two middle digits (Y's) can take 10 values (0, 1, ..., 9), so total such numbers possible is 5*10=50.

Hope it's clear.

Hi Bunuel..

Can u tell me why we r doing 5*10 ways..

Because of Principle of Multiplication.

Principle of Multiplication
If an operation can be performed in ‘m’ ways and when it has been performed in any of these ways, a second operation that can be performed in ‘n’ ways then these two operations can be performed one after the other in ‘m*n’ ways.
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Re: A palindrome is a number that reads the same forward and  [#permalink]

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A 4 digit palindrome is of the form _ _ / _ _ where each _ represents a digit and / represents that the 1st and 2nd _'s are mirror images of 4th and 3rd _'s respectively.
Now let's find how many numbers can satisfy the dashes.
Let's start from right to left
4th _ - since it's an odd number we have 1, 3, 5, 7, 9 -> 5 different digits
3rd _ - since you can have all 10 digits - 0,1,2,3,4,5,6,7,8,9 -> 10 different digits
Now remember for the number to be a palindrome, the 1st and 2nd digits have to match 4th and 3rd digits. Hence you can have only one number for each 1st and 2nd number.
Hence the solution is - 1 * 1 * 10 * 5 = 50

Sent from my iPhone using GMAT Club Forum Re: A palindrome is a number that reads the same forward and   [#permalink] 02 Nov 2017, 00:04

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