Palindrome 5 digit

[yass20015]

A palindrome is a number that reads the same forward and backward. For example, 2442 and 111 are palindromes. If 5 digits palindromes are formed using one or more of the digits 1, 2 and 3 , how many such palindromes are possible ?
a) 12
b) 15
c) 18
d) 24
e) 27

What do they mean by "using one or more" of the digits ?
I solve lit like this: XYZYX
3*3*3*1*1 = 27
is it ok ?
Thanks

[GMATGuruNY]

Your solution looks perfect.

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) 24
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.

[Brent@GMATPrepNow]

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) 24
E) 27

Take the task of building palindromes and break it into stages.

Stage 1: Select the ten-thousands digit
We can choose 1, 2, or 3
So, we can complete stage 1 in 3 ways

Stage 2: Select the thousands digit
We can choose 1, 2, or 3
So, we can complete stage 2 in 3 ways

Stage 3: Select the hundreds digit
We can choose 1, 2, or 3
So, we can complete stage 3 in 3 ways

IMPORTANT: At this point, the remaining digits are already locked in.

Stage 4: Select the tens digit
This digit must be the SAME as the thousands digit (which we already chose in stage 2)
So, we can complete this stage in 1 way.

Stage 5: Select the units digit
This digit must be the SAME as the ten-thousands digit (which we already chose in stage 1)
So, we can complete this stage in 1 way.

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

Answer: E
--------------------------

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Cheers,
Brent

[Brent@GMATPrepNow]

Here's another (slightly trickier) palindrome question to practice with: https://www.beatthegmat.com/palindrome-t263705.html


And here's a palindrome question I created for BTG a longggg time ago (for their Math Challenge Question contest):

A palindrome is a word that is read the same backwards as forwards. For example, the words "BADAB," "IAGAI," and "HHHHH" are all palindromes.

How many 5-letter palindromes can be created using the letters A, B, C, D, E, F, G, H, I and J?

For a full solution, watch the following YouTube video: https://youtu.be/qfiPnXIBx7g

Cheers,
Brent

[[email protected]]

Hi yass20015,

Your solution is correct.

To answer your question - the phrase 'using one or more OF the digits 1, 2 and 3' means that the palindrome can use any combination of those numbers.

For example, we could have....
11111
12121
33233
32123
Etc.

GMAT assassins aren't born, they're made,
Rich

[Max@Math Revolution]

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.


OK you just need to decide the first three digits, and "using one or more of" of the digits are considered repetitions. Thus, 3*3*3*1*1 = 27 is the answer.


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[Matt@VeritasPrep]

What do they mean by "using one or more" of the digits ?

They mean you can repeat the digits if you like. For instance, if you only use one of them, all of the digits would be the same, and you would have 11111, 22222, or 33333. If you only use two of them, you'd choose two of the three numbers, then place them in some palindromic order: 11211, or 21212, or 13131, etc. etc.

[gmatbeater1989]

A palindrome is a number that reads the same forward and backward. For example, 2442 and 111 are palindromes. If 5 digits palindromes are formed using one or more of the digits 1, 2 and 3 , how many such palindromes are possible ?
a) 12
b) 15
c) 18
d) 24
e) 27

All we need to do is figure out how many ways to set the first 3 digits (since the last 2 are forced to be the same as the first 2)

List:
111
112
113
121
122
123
131
132
133
211
212
213
221
222
223
231
232
233
311
312
313
321
322
323
331
332
333

Count them up to get 27

[[email protected]]

Hi gmatbeater1989,

Using your same logic, since each of the 3 digits could be any of the 3 options, then we have (3)(3)(3) = 27 options.

GMAT assassins aren't born, they're made,
Rich

[Matt@VeritasPrep]

All we need to do is figure out how many ways to set the first 3 digits (since the last 2 are forced to be the same as the first 2)

You could also just set the first two, then realize that each of them has three options that goes with it.

For instance, suppose I start with 11. I could finish with 1, 2, or 3 (111, 112, 113), so there are three options beginning 11.

Since I can begin 11, 12, 13, 21, 22, 23, 31, 32, 33, I have 9 initial codes with 3 endings each, for a total of 27.

Of course, you can go even further and notice that you have 3 options for each space, so 3 * 3 * 3 = 27 is your answer.