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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 Updated on: 24 Jul 2019, 10:04
Take the palindrome as ABCACBA

A = 5 or 6 -->> 2 possibilities
B = 5 or 6 or 0 -->> 3 possibilities
C = 5 or 6 or 0 -->> 3 possibilities
Middle single A = 5 or 6 or 0 -->> 3 possibilities

So total numbers = 2*3*3*3 = 54

Corrected the mistake and reposted
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Originally posted by Jiggy11 on 22 Jul 2019, 12:05.
Last edited by Jiggy11 on 24 Jul 2019, 10:04, 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, 12:16
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?

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

To form a 7 digit number, the first digit can't be 0, so only 2 ways to choose the 1st digit - 5 or 6.
2nd digit could be one of 5, 6, 0 - hence 3 ways to choose
3rd digit also could be one of 5,6,0 - hence 3 ways to choose
4th digit also could be one of 5, 6, 0 - hence 3 ways to choose.

The remaining digits are all a mirror of the 1st three, hence only 1 way to choose.

So, final answer = 2*3*3*3*1*1*1 = 54.


Correct answer B
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New post 22 Jul 2019, 12:36
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A seven-digit palindrome: \(ABSDSBA\)

We can use any of or all of the following digits: \(5, 6, 0\)

Examples of palindrom: \(5555555, 6665666, 5006005,\) but not \(0560650\). The last one is a six-digit number, so we can't use \(0\) as the starting digit.

Correspondingly, we have \(2\) choices (\(5\) and \(6\)) for \(A\), and by \(3\) choices (\(5,6,\) and \(0\)) for each of \(B, S,\) and \(D\).

The number of possible palindromes: \(2*3*3*3=54\)

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, 12:47
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?

Let's say that _ is missing digit:
_ _ _ _ _ _ _ - 7 digits
And there are 3 possible variants of digit: 5,6,0
So the first digit can be 5 and 6 (if 0 it's not 7 digit number)
2*3*3*3*1*1*1 = 54
Why is the last 3 digits 1 ?
Because this number is a palindrome, that has the same end as the beginning.

ANSWER B

A. 16
B. 54
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]

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New post 22 Jul 2019, 12:52
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IMO Correct answer is B;

Explanation is provided as attachment
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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, 13:03
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7 digit palindrome would be like ABCDCBA.
So we have for A: two options: 5 or 6
rest B,C &Dcan be any of the 3 options. therefore answer is 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, 13:28
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?

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


The question looks scary, especially when we see the range of the given answers.
But hold on, it is not so hard!
First of all, let's imagine the possible 7-digit palindromes. It is clear that we are interested only in first 4 digits, because the last 3 digits will copy the first 3.
Besides, we know that the first digit can't be zero.
So we have 2 possible options (5 and 6) for the first digit and 3 options (5, 6, 0) for each of the remaining 3 digits.
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, 15:02
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The digits to choose from to create palindromes are 5,6, and 0.
Since we are to create 7 figure palindromes, there is a restriction on position 1 from the available digits. 0 cannot be the first digit and by extension, it cannot be the last digit. Therefore, we are limited to the selection of either 5 or 6 and there are 2C1 possible ways of selecting either for the first and seventh digit.

Second, third and fourth digit do not have any restrictions. Therefore there are 3C1 ways of selecting a number among the three digits to fill those positions.

The total number of possible palindromes that can be created is therefore given as:
2C1*3C1*3C1*3C1=2*3*3*3=54.
The answer is therefore 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, 16:35
Given: there are 3 number choices (5,6,0)

For 7th digit to be palindrome, 1-7, 2-6, 3-5 must be same number.
4th digit can be any number.

so there are 3 choices for pair (1-7), pair (2-6), pair (3-5) and (4)

thus 3*3*3*3 = 3^4 = 81.

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 22 Jul 2019, 16:38
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Any 7-digit palindromes can be written as 'abcdcba'.

Number of values that a can take= 2 (5 or 6)

Number of values that b, c and d can take= 3 (0, 5 or 6)

Total number of 7-digit palindromes using 0, 5 and 6 = 2*3*3*3*1*1*1= 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, 17:23
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Palindrome of 7 digits -> XYZPZYX
Digits to choose from - 5, 6, 0
how many palindromes are possible ?
As we can see we have to find only X, Y, Z, P -> combinations in Mix
first digit cannot be 0 ; X can be 5 or 6
X = 2 ways (5, 6)
Y = 3 ways (5, 6, 0)
Z = 3 ways (5, 6, 0)
P = 3 ways (5, 6, 0)

total ways = 2 * 3 * 3 * 3 = 54 ways

B is the answer!
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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, 18:15
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Let the 7-digit number be a b c d c b a.

So, we only need to find out how many we can fill a, b, c and d, because the last three digits take the same number.
In addition to that, all a, b, c, and d have to be of the digits 5, 6 and 0 (repetition is allowed)

• a can take \(2\) numbers, which are 5 and 6. 0 is definitely out.
• b can take \(3\) numbers, which are 0, 5 and 6.
• c can take \(3\) numbers, which are 0, 5 and 6.
• d can take \(3\) numbers, which are 0, 5 and 6.

Thus, way to form a 7-digit palindrome number formed using one or more of the digits 5, 6 and 0 = \(2 * 3 * 3 * 3 * 1 * 1 * 1= 54\).

Answer is option (B) \(- 54\)
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New post 22 Jul 2019, 19:13
Answer is 81.

This is because while it is a 7 letter combination, you actually only have 4 slots to fill because the first and last is the same (1st slot), 2nd and 6th (2nd slot), 3rd and 5th (3rd slot) and finally the 4th.

So because you have 3 numbers to choose from with replacement, the choices are
3 x 3 x 3 x 3

Ie 81

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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, 19:17
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the number can be constructed from 3 digits 5,6,0. Zero cannot be in first digit place. So only 2 ways of arranging the first digit. the middle point is digit 4. Digit 2 to digit 4 can be arranged in 3 ways each. The remaining digits can only be arranged in one way since this is a palindrome.

Total possible ways= 2*3*3*3*1*1*1=54 ways
see the picture.

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New post 22 Jul 2019, 19:35
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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?

Make 7 slots
_ _ _ _ _ _ _
There are 2 possibilities of the first slot (the number cannot start with 0 as this would form a 6 digit number)
2 _ _ _ _ _ _
There are 3 possibilities of the second slot
2 3 _ _ _ _ _
Similar to the second lost, there are 3 possibilities of the third slot and of the fourth slot
2 3 3 3 _ _ _
In order for it to be a palindrome, the fifth slot must be the same as the third slot, 1 possibility
2 3 3 3 1 _ _
Similar logic follows for the last two slots (there can only be one possibility for each for it to be a palindrome)
2 3 3 3 1 1 1
Multiply all together
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, 19:52
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7 digit palindrome mean it should read the same from 4th digit.
Number of ways 1st digit can be filled=2 (since we can't use 0)
2nd digit=3 ways
3rd digit=3 ways
4th digit=3 ways
All digits after this can be filled only in 1 way
Therefore, total possible palindromes= 2*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, 20:01
1
XABCBAX

-- X can have 2 values
-- A can have 3 values
-- B can have 3 values
-- C can have 3 values

=2*( 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, 20:24
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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?

_ _ _ _ _ _ _
For a palindrome 1st and last should be same. similarly 2nd and 2nd last should be same.
There is only one choice for last as it has be same as 1st
1st can only be 2 as 0 not allowed
ways to fill 1st and last will be 2 _ _ _ _ _ _ 1
2nd and 2nd last will be 2 *3 _ _ _ _ 1* 1
2 *3*3*3*1*1*1
thus answer 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, 20:29
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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

Sol:

A 7-digit palindrome will be of the form
- - - a - - -
The first three digits is what wee need to decide, as the last three digits have to be repeated to make a palindrome.

For the first digit, we have 2 choices (5 or 6). We can't start the number with 0 as the first digit.
For the second digits, we have 3 choices (0, 5 or 6). The digits can be repeated.
For the third digit, we have 3 choices (0, 5 or 6). The digits can be repeated.
For a, we have 3 choices (0, 5 or 6)

Hence, total palindromes that can be made = 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, 20:34
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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?

7 digit palindrom = A B C D C B A = 2*3*3*3 *1*1*1
A can can 2 values - 5,6 =2
B can take 3 values = 5,6,0 =3
C can take 3 values = 5,6,0 = 3
D can take 3 values = 5,6,0 =3
Next C B A will take only 1 way following the first A B C

Total = 2*3*3*3 *1*1*1 = 27*2 = 54
- Option B
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Re: A palindrome is a number that reads the same forward and backward. For   [#permalink] 22 Jul 2019, 20:34

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