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A palindrome is a number that reads the same forward and backward. For

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

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New post 22 Jul 2019, 08:00
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A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?

A. 16
B. 54
C. 81
D. 486
E. 729

 

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

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New post Updated on: 23 Jul 2019, 00:39
2
We have been given three digits – 5, 6 and 0 to form a seven-digit number.
Now, we cannot have a seven-digit number with a 0 as the first digit, so we only have 5 and 6 available as the first digit.

The number of ways to fill first four digits of the palindromic seven-digit number is 2 * 3 * 3
The middle digit can be any digit, so it can be filled in 3 ways
The last four digits will repeat the same digit that has occurred in the first four digits (to form the palindrome). Therefore, they can be formed in 1 * 1 * 1 ways

Thus, the number of such palindromes possible are 2 * 3 * 3 * 3 * 1 * 1 * 1 = 54 ways.

Answer B

Originally posted by Sayon on 22 Jul 2019, 08:14.
Last edited by Sayon on 23 Jul 2019, 00:39, edited 1 time in total.
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A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post Updated on: 22 Jul 2019, 09:02
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A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?

A. 16
B. 54
C. 81
D. 486
E. 729

Given: A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0.
Asked: How many such palindromes are possible?

__ __ __ ___ __ __ __ __

For 1st digit, 2 choices 5 & 6 are available
For 2nd digit, 3 choices 5,6 & 0 are available
For 3rd digit, 3 choices 5,6 & 0 are available
For 4th digit, 3 choices 5,6 & 0 are available
For 5th digit, only 1 choice is available
For 6th digit, only 1 choice is available
For 7th digit, only 1 choice is available

Total number of palindromes = 2*3*3*3*1*1*1 = 54

IMO B
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Originally posted by Kinshook on 22 Jul 2019, 08:15.
Last edited by Kinshook on 22 Jul 2019, 09:02, edited 1 time in total.
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:18
1
IMO answer is B:

ABCXCBA --> X can be A or B or C
first place A can take 5 or 6 - 2 choices
second place B can take 0,5 or 6, 3 choices
similarly for C 3 choices

total = 2*3*3 = 18, but this 18 is possible for 1 x value. for 3 x values. 3*18 =54
so B
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:19
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Consider 7 spots.

_ _ _ _ _ _ _
0 0 0
5 5 5 5
6 6 6 6

First blank can be filled in 2 ways -5/6
2nd can be filled in 3 ways-0/5/6
3 & 4 also 3 ways.

the last 3 blanks depends on the first three so they can be filled in only 1 ways..
total - 2*3*3*3
=54
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:27
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We have to form a seven-digit palindrome. Which means if we divide it in two parks, both mirror each other. In essence we only need to find the first 4 digits, the rest will be just the mirrors.
1st digit cannot be 0.
No. of ways to fill
1st slot - 2
2nd slot - 3
3rd slot 3
4th slot -4
Therefore, Total no of ways - 2*3*3*3 = 54
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:28
1
B

We need make palindromes of 7 digits with 3 digits that are 5, 6, 0.

Lets start filling this 7 digit palindrome from the middle.

The middle digit can be filled in 3 different ways with either of 5, 6 or 0. Now, either side of the middle digit will take same number out of 5, 6 or 0.

So, digit immediately besides can also be filled in 3 different ways, then the second and sixth place can also be filled in 3 different ways, but the extremes can be filled only in 2 ways since 0 cant go on the left end, otherwise it will be 6 digit no.

So, total possible ways: 3*3*3*2 = 54
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:32
Using the placement method:

_ _ _ _ _ _ _

1st spot in 2 ways (5 and 6, since it has to be a 7-digit number we cannot have 0 as the first digit)
2nd and 3rd spots in 3 ways (5,6,0)

Corresponding to the 1st, 2nd and 3rd spots the 7th, 6th and 5th spots can be filled in 1 way each.

and the middle/4th spot can again be filled in 3 ways (5,6,0)
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A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:32
1
A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?


In 7-digit palindrome, we will have 3 repeated digits.
So basically we are asked how many 4- digit numbers we can create using just 3 digits - 5,6 and 0
since zero cannot be the first number, we can conclude
2*3*3*3=54

The answer is B
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:40
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1st digit has only 2 option (5,6) =2
7th digit has to be same as 1st = 1
2nd digit has only 3 option (5,6,0) =3
6th digit has to be same as 2nd = 1
3rd digit has only 3 option (5,6,0) =3
5th digit has to be same as 3rd = 1
4th digit has only 3 option (5,6,0) =3

THerefore, total possible combinations=2*1*3*1*3*1*3=54

Hence B
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:40
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A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?

This is a medium level difficulty problem. It is a combination of probability and number property.

We know that out of the 7 Digits 0 can not be the first digit.
So. 2*3*3*3*1*1*1 = 54

A. 16
B. 54
C. 81
D. 486
E. 729

Hence the answer is B.
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:41
Let The palindrome be

ABCDCBA

If we select 1st four digits i.e A,B,C,D then for the last 3 only 1 possible option is available.

A --> 3 ways
B-> 3 ways
C --> 3 ways
D -->3 ways

Total no of ways = 3X3X3X3 =81

Hence, C is the answer.
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New post Updated on: 22 Jul 2019, 19:33
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palindrome number is: PQTRTQP

P can be 5 or 6 -> 2 options
Q can be 0, 5, or 6 -> 3 options
T can be 0, 5, or 6 -> 3 options
R can be 0, 5, or 6 -> 3 options

S, # of ways is 2*3*3*3= 54

(B) is our answer.

Originally posted by Mizar18 on 22 Jul 2019, 08:44.
Last edited by Mizar18 on 22 Jul 2019, 19:33, edited 1 time in total.
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A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post Updated on: 22 Jul 2019, 10:31
1
Quote:
A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?

A. 16
B. 54
C. 81
D. 486
E. 729

7 digits ==> __ __ __ __ __ __ __
1 2 3 4 5 6 7
As its a palindrome, digits 5,6,7 will be the same as 3,2,1 respectively. So we don't need to consider them.

As 7 digit number, 0 cant be the 1st digit.
Hence 1st digit = 2 possibilities
2nd digit= 3 possibilities
3rd digit = 3 possibilities
4th digit = 3 possibilities
Hence total no. of ways,
2 * 3 * 3 * 3 = 54
Option B is the answer

Originally posted by techloverforever on 22 Jul 2019, 08:45.
Last edited by techloverforever on 22 Jul 2019, 10:31, edited 1 time in total.
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:49
2
Quote:
A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?

A. 16
B. 54
C. 81
D. 486
E. 729


number of options: {5,6,0} = 3
7-digit palindrome is basically: ABCDCBA
[A] cannot be 0, so 2 choices
[B] can have any 3
[C] any 3
[D] any 3
The duplicate letters need to be the same as the former, so total: 2*3*3*3=2*27=54

Answer (B).
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:50
2
Total digits to be used : 5,6,0
7 digit palindrome can be formed with the first digit being either 5 or 6

Identify the number of possibilities to assign a number across 7 digit places

1st digit - highest- can be filled by (5 or 6) - No of ways :2
7th digit - last digit - should be filled by the same number for palindrome, so no of ways :1
2nd digit - from highest digit- can be filled by (5 or 6 or 0) - No of ways :3
6th digit - from last digit - should be filled by the same number for palindrome, so no of ways :1
3rd digit - from highest digit- can be filled by (5 or 6 or 0) - No of ways :3
5th digit - from last digit - should be filled by the same number for palindrome, so no of ways :1
4th digit - middle digit - can be filled by (5 or 6 or 0) - No of ways :3

So total possible numbers : 2*3*3*3 = 54
IMO :B
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:51
A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?

A. 16
B. 54
C. 81
D. 486
E. 729

5,6,0 can be arranged in xyzzzyx , xyyzyyx , yxxzxxy or say arrangement is possible in 3*2*1 * 3 ; 18 ways
and xyz can be arranged ; 3^3 ; 27 ways
so total combinations ; 27* 18 ; 486
IMO D
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:53
IMO C

A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?

A. 16
B. 54
C. 81
D. 486
E. 729

Sol:

this has to be a 7 digit number, so we know that there are three numbers to choose from

we have 3 choices for the first 4 numbers and then the last three numbers has to be the same number which was already choose for 1st place, 2nd place and 3rd place respectively.

for the fourth place we can choose any of the three without any obligation

so 3*3*3*3*1*1*1* =81

C is correct.
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 08:59
2
IMO : B
This is a combination problem . As this involves one no. 0, so we can use technique, All- Minus no. with Zero,: As when zero will be first no. it cannot be 7 digit, it will become 6 digit which is the trap here.
Technique to solve palindrome questions :
Consider no. are like : abcdabc , or abababa, or bababab : You can see that we just need to see first 4 digit, rest 3 are just image of first 3 :
And 4th no. will be same as always if we come forward or backward .
So we have no. for place 1-4 : 3*3*3*3 *1*1*1 : 81 :
For image place 5,6,7th , we take it 1 only,

Now 81 is plaindrome using all no.
Considering 0 on first place we get : 1*3*3*3*1*1*1 = 27 :
81-27 = 54 :
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Re: A palindrome is a number that reads the same forward and backward. For  [#permalink]

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New post 22 Jul 2019, 09:02
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A palindrome is a number that reads the same forward and backward. For example, 3663 and 23232 are palindromes. If 7-digit palindromes are formed using one or more of the digits 5, 6 and 0, how many such palindromes are possible?
Solution:
7 digit palindrome: so
=> 1st digit = 7th digit = 5/6 but not 0(if zero it will be 6 digit number) --> 2 choices
=> 2nd digit = 6th digit = 5/6/0 --> 3 choices
=> 3rd digit = 5th digit = 5/6/0 --> 3 choices
=> 4th digit = 5/6/0 --> 3 choices
So total number of choices = 2*3*3*3=54

A. 16
B. 54 --> correct
C. 81
D. 486
E. 729
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Re: A palindrome is a number that reads the same forward and backward. For   [#permalink] 22 Jul 2019, 09:02

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