A palindrome is a number that reads the same forward and backward. For example, 2442 and 111 are palindromes...

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A palindrome is a number that reads the same forward and backward. For example, 2442 and 111 are palindromes. If 5-digit palindromes are formed using one or more of the digits 1, 2, 3, how many such palindromes are possible?

A. 12
B. 15
C. 18
D. 28
E. 27

OA E

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AAPL wrote:
Mon Feb 22, 2021 10:13 am
GMAT Prep

A palindrome is a number that reads the same forward and backward. For example, 2442 and 111 are palindromes. If 5-digit palindromes are formed using one or more of the digits 1, 2, 3, how many such palindromes are possible?

A. 12
B. 15
C. 18
D. 28
E. 27
To read the same forward and backward, the 5-digit integer must look as follows:
ABCBA.
The ten-thousands digit and the units digit must be THE SAME.
The thousands digit and the tens digit must also be THE SAME.

Number of options for the ten-thousands digit = 3. (1, 2, or 3)
Number of options for the units digit = 1. (Must be the same as the ten-thousands digit)
Number of options for the thousands digit = 3. (1, 2, or 3)
Number of options for the tens digit = 1. (Must be the same as the thousands digit)
Number of options for the hundreds digit = 3. (1, 2, or 3)
To combine these options, we multiply:
3*3*3*1*1 = 27.

The correct answer is E.
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AAPL wrote:
Mon Feb 22, 2021 10:13 am
GMAT Prep

A palindrome is a number that reads the same forward and backward. For example, 2442 and 111 are palindromes. If 5-digit palindromes are formed using one or more of the digits 1, 2, 3, how many such palindromes are possible?

A. 12
B. 15
C. 18
D. 28
E. 27

OA E

Solution:

We have 3 options for the first digit, 3 options for the second, 3 options for the third, 1 option for the fourth (since it has to be the same as the second), and 1 option for the fifth (since it has to be the same as the first). Thus, there are 3 x 3 x 3 = 27 possible palindromes.

Answer: E

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AAPL wrote:
Mon Feb 22, 2021 10:13 am
GMAT Prep

A palindrome is a number that reads the same forward and backward. For example, 2442 and 111 are palindromes. If 5-digit palindromes are formed using one or more of the digits 1, 2, 3, how many such palindromes are possible?

A. 12
B. 15
C. 18
D. 28
E. 27

OA E
Take the task of creating a 5-digit palindrome and break it into stages.

Stage 1: Select a digit for the first position.
We can choose 1, 2 or 3, so we can complete stage 1 in 3 ways

Stage 2: Select a digit for the second position.
We can choose 1, 2 or 3, so we can complete stage 2 in 3 ways

Stage 3: Select a digit for the third position.
We can choose 1, 2 or 3, so we can complete stage 3 in 3 ways

Stage 4: Select a digit for the fourth position.
Important: In order to create a palindrome, the fourth digit must be the same as the second digit.
For example, if the first three digits are 213, then fourth digit must be 1, and the fifth digit must be 2 to get the 5-digit palindrome 21312

Since the fourth digit must be the same as the second digit, we can complete stage 4 in 1 way

Stage 5: Select a digit for the fifth position.
Since the fifth digit must be the same as the first digit, we can complete stage 5 in 1 way

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus create a 5-digit palindrome) in (3)(3)(3)(1)(1) ways (= 27 ways)

Answer: E
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