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

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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, 20:55
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Question asks to find the number of 7-digit palindromes formed using one or more digits 5, 6 and 0.

It means the three digits have to be repeated maximum of 7 times(except '0') and minimum of 1.

Now here's how i approached. We have seven positions to fill using digits 5, 6 and 0.
1 2 3 4 5 6 7
_ _ _ _ _ _ _

For positions ‘5’, ‘6’ and ‘7’ we have only one possibility for each as they would be equal to digits at positions ‘3’, ‘2’ and ‘1’ respectively.

For position ‘1’ we have two possibilities i.e. digits 5 and 6 can be placed here since placing ‘0’ would invalidate 7-digits palindromes.

Similarly, for position ‘2’ , ‘3’ and ‘4’ we have three possibilities i.e. digits 5, 6 and 0 can be placed here.

Hence number of possible 7-digit palindromes are:
2x3X3x3x1x1x1 = 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, 21:03
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For combinatorics questions, I like to use so called slots method. We need to create a seven digits number that reads back and forth the same. For example, 1114111 OR 1234321. Back to questions, for out 7-digit number, we can use only 5,6,and 0. So, our number should look like this: ABCDCBA.
Slots method: _ _ _ _ _ _ _
For first slot, we have two options 5 or 6 (0 cannot be first integer because then we will have 6 digits). Since last digit should be reflective of the first digit, for units digit, we have only one option
For second slot, we have 3 options. For second slot from the end, we have one option since it must be reflective of 2nd digit from beginning.
For third slot, we again have 3 options. For 3rd digit from end, we have one option.
For fourth slot, we again have 3 options. Now, we need to multiply all options to get total number of different combinations of 7 digit number.
2*3*3*3*1*1*1= 54 (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, 21:54
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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?

_ _ _ _ _ _ _
The first digit (left most) can only have value 5 or 6. i.e. 2 values
Second digit can have any value 5 or 6 or 0. (since digits can be repeated). i.e. 3 values
Similarly 3rd digit can have 3 values.
4th can have 3 values.
Now that the number is palindrome.
The 5th digit will be same as 3rd digit. Hence can take only 1 value.
similarly 6th and 7th digit can also take only 1 value.

Therefore total possible palindromes will be 2*3*3*3*1*1*1 = 54

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

Answer Choice: 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, 22:20
1
Given,

All the digits of a 7-digit palindrome are from 5,6,0 with repetitions.

To find,

Number of such palindromes possible.

Let the 7-digit number be depicted as - _ _ _ | _ | _ _ _

Approach 1: Probability approach -

Number of ways in which we can arrange the first digit = 2 (Since it can only be 5 or 6).
Number of ways in which we can arrange the second digit = 3 (Since it can be 5 or 6 or 0).
Number of ways in which we can arrange the Third digit = 3 (Since it can be 5 or 6 or 0).
Number of ways in which we can arrange the Fourth digit = 3 (Since it can be 5 or 6 or 0).
Number of ways in which we can arrange the Fifth digit = 1 (Since it has to be the same as the ten thousands digit to make it a palindrome)
Number of ways in which we can arrange the Sixth digit = 1 (Since it has to be the same as the one lakh's digit to make it a palindrome)
Number of ways in which we can arrange the Seventh digit = 1 (Since it has to be the same as the ten lakh's digit to make it a palindrome)

Now,

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

Approach 2: Combinatorics approach -

Number of ways in which we can select the first digit = 2C1 = 2 (Since it can only be from 5 or 6).
Number of ways in which we can select the second digit = 3C1 = 3 (Since it can only be from 5 or 6 or 0).
Number of ways in which we can select the Third digit = 3C1 = 3(Since it can only be from 5 or 6 or 0).
Number of ways in which we can select the Fourth digit = 3C1 = 3(Since it can only be from 5 or 6 or 0).
Number of ways in which we can select the Fifth digit = 1C1 = 1 (Since it has to be the same as the ten thousands digit to make it a palindrome)
Number of ways in which we can select the Sixth digit = 1C1 = 1 (Since it has to be the same as the one lakh's digit to make it a palindrome)
Number of ways in which we can select the Seventh digit = 1C1 = 1 (Since it has to be the same as the ten lakh's digit to make it a palindrome)

Now,

Total number of possible palindromes = 2 * 3 * 3 * 3 * 1 * 1 * 1 = 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, 23:51
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?

First digit can be 5 or 6 - 2 ways.
Second digit can be 5, 6 or 0 - 3 ways
Third digit can be 5, 6 or 0 - 3 ways
Fourth digit can be 5, 6 or 0 - 3 ways
Fifth, sixth and seventh digits should be same as third, second and first digits respectively.

No of ways = 2*3*3*3*1*1*1 = 54.

Option B.
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New post 23 Jul 2019, 00:07
1
Answer should be B

Since the last 3 digits would be same as first 3
Hence the 1st digit can have 2 option
2nd 3
3rd 3
4th 3
Hence possible palindromes = 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 23 Jul 2019, 00:27
Digits are 5,6,0
We need to make 7 digits palindrome:
This can be made by:
Using 5,6,0 for first three digits: such as 056,065,506,560,605,650 Total 6 combination.
For the middle digit, we can use, 0,5,6 for one combination: 0560,0565,0566
For the last three digits, the number will be 1 as those three digits will be the same as the first one.
Now, if we take only two digits:
0,5 : 000,005,050,055,555,550,505,500 : Total 8 combination
For middle digit: 2 (0,5)
And 0,5 : 05,50 : 0505050 or 5050505
0,6 : 0606060 or 6060606
5,6: 5656565 or 6565656

Now if we take only one digit:
0000000,5555555,6666666 : Total 3 combination

So, total possible values: 81

IMO the answer is C.

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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 23 Jul 2019, 00:56
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?
B. 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 23 Jul 2019, 01:20
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?

First digit will be able to take only 2 values 5 & 6.
First digit and last digit will be equal so so the last digit will also have same value as first.

Similarly, second digit and second last digit will have equal value. Same for the 3rd digit and third last.

Second and second last digit can have 3 values. Similarly, 3rd digit can have 3 values and 4 th digit can have 3 values.

So we can say the arrangements can be:
2*3*3*3 = 54

Answer: B 54

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New post 23 Jul 2019, 02:23
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since the number is a palindrome number, we need to find the number of ways of filling the number upto the middle digit of the number (here it is 04 as total digit are given to be 07)
required 07 digit number = D1 D2 D3 D4 D5 D6 D7
here, D1 is the 1st digit, D2 is the 2nd digit and so on...
D1 - can be filled by 2 ways i.e. 5 or 6 (here D1 cannot be filled by 0 as the number will no longer be a 7 digit number then)
D2- can be filled by 3 ways (5,6,0)
D3- can be filled by 3 ways (5,6,0)
D4- can be filled by 3 ways (5,6,0)
D5- can be filled by 01 way (As it a palindrome number, this D5 will be same as D3)
D6- can be filled by 01 way (reason same as mentioned in D5)
D7- can be filled by 01 way (reason same as mentioned in D5)
therefore, total number of ways = 2x 3x3x3x1x1x1 = 54 palindrome numbers can be formed

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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, 12:35
Question: 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?

Seven digit number...
Slot #1 Options: 2 (5 and 6)
Slot #2 Options: 3 (5, 6, or 0)
Slot #3 Options: 3 (5, 6, or 0)
Slot #4 Options: 3 (5, 6, or 0... since slot #4 isn't mirrored it can be any of the three available digits 5, 6, and 0.)
Slot #5 Options: 1 (Mirrored option #3 choice)
Slot #6 Options: 1 (Mirrored option #2 choice)
Slot #7 Options: 1 (Mirrored option #1 choice)

\(=\)(2)(3)(3)(3)(1)(1)(1)
\(=\)(54)(1)
Correct Answer: B. 54
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Originally posted by duchessjs on 23 Jul 2019, 02:26.
Last edited by duchessjs on 23 Jul 2019, 12:35, 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 23 Jul 2019, 02:58
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Since we need to form a 7 digit palindrome let us consider the middle most digit i.e at the 4th position it need to be fixed there fore it is an arrangement question so arranging 1 number out of 3 in 3 ways now let us consider the first digit ,it cannot be zero so it can be arranged in 2 ways similarly 2 and 3 position can be done in 3 ways each.We need to consider the first 4 number only because the rest will be repeated therefore it can be done In =2*3*3*3
=54 ways
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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 23 Jul 2019, 03:10
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Solution:

Let the digits of the seven digit number be ABCDEFG , Since this number is a palindrome, we know that the first 3 digits and the last 3 digits must be similar, the middle number however; can be any number of the given options as it will be the last number from front and from the back.
So accordingly,
A will have 2 choices i.e 5,6 . It is important to note here that since it is a 7 digit number, it cannot start with a zero so zero can't be a choice here
B will have 3 choices. i.e 5,6,0
C will have 3 choices. i.e 5,6,0
Last 3 options must be similar to what A B & C is, therefore digits E,F,G will have only one choice
D, the middle one, will also have 3 choices i.e 5,6 & 0
Since it's a combination question, we must multiply the choices, we get 2 X 3 X 3 X 3 X 1 X 1 X 1 i.e 54.

Hence the answer must be 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 23 Jul 2019, 03:13
need: abcdcba
2 options for a (5 and 6)
3 options for b - (0, 5, and 6)
3 options for c - (0, 5, and 6)
3 options for d - (0, 5, and 6)
1 option for c after d (same digit used for c before d)
1 option for b after d (same digit used for b before d)
1 option for a after d (same digit used for a before d)
Overall: 2*3*3*3*1*1*1=54
Answer D
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New post 23 Jul 2019, 03:39
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Answer is B

For a 7-digit palindrome,

Since we can use 5,6, and 0,

For the 1st place we have 2 options and for the last place we have 1 option [We cannot use 0 in the first place]
For the 2nd place we have 3 options and for the 6th place we have 1 option
For the 3rd place we have 3 options and for the 5th place we have 1 option
For the 4th place we have 3 options

Now multiply - 2*3*3*3*1*1*1 = 54
Therefore if we form 7-digit palindromes by using one or more of the digits 5, 6 and 0 then we can make 54 such palindromes are possible.

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 23 Jul 2019, 04:02
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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

Solution:
As there is a zero also to be considered among the numbers to form a 7 digit number ,the first digit will always have 2 possibilities (as zero acnnot occupy ,else then it will not be a seven digit number).Aslo 5th ,6th,and 7th digit will replicate the first three.Hnece following will be the arrangements.


2 P1 3p1 3p1 3p1 ( 1 ) (1) ( 1) =2*3*3*3=54 ways
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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 23 Jul 2019, 04:38
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

As all of the option seems like square value, then 54 should be accurate. correct answer choice should be 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 23 Jul 2019, 04:40
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We got only 3 digits to form the 7-digit palindromes. (5,6 and 0):

For the 1st digit, there are 2 options(5 or 6, the number cannot start with zero)
For the 2nd digit, there are 3 options (5,6,0)
For the 3rd digit, there are 3 options (5,6,0)
For the 4th digit, there are 3 options too (5,6,0) (Because, the 4th digit is in the middle of 7-digit number and )

the last 3 digits depend on the first 3 digits:
--> the 5th digit should be the same as 3rd digit;
--> the 6th digit should be the same as 2nd digit;
--> the 7th digit should be the same as 1st digit.
They all have just one option in every palindromes

Well, we need to multiply all options:
--> 2*3*3*3*1*1*1=54 palindromes.

The answer choice 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 23 Jul 2019, 05:20
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IMO B.

A 7 digit number which needs to be a palindrome. 5,6 and 0 can be the digits of the number.
Lets fill in the dashes-

1 2 3 4 5 6 7

1. Can be filled with either 5 or 6 -> 2 ways.
2. Can be filled with any- 5,6 or 0 --> 3 ways.
3. Can be filled with any- 5,6 or 0 --> 3 ways.
4. Can be filled with any- 5,6 or 0 --> 3 ways.
5. Can be filled with only one way, the same number as 3.
6. Can be filled with only one way, the same number as 2.
7. Can be filled with only one way, the same number as 1.

Hence, total ways = 2*3*3*3 = 54 ways.
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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 23 Jul 2019, 05:39
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so there are 7 digits out of which the first cannot be 0.

The first digit can be essentially filled with 2 digits.
the second,third and fourth and can be filled in 3 ways.
Now for this number to be a palindrome we must ensure that the 2nd digit=6th and 3rd=5th.

so the total number of possibilities=2*3*3*3*1*1*1=54(IMO the correct option is B)
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Re: A palindrome is a number that reads the same forward and backward. For   [#permalink] 23 Jul 2019, 05:39

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