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? GMAT Explanation + Additional Examples!
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?
Define the question: How many such palindromes are possible?
As usual, start at the end by defining the question.
Another way of stating this is — how many combinations of numbers are possible given the conditions:
- 5 digits
- Digits must be 1, 2, or 3.
Ready for some combinatorics?
Organize: Let’s break down palindromes and see if we can spot a pattern.
First off, we know our palindromes are five digits, so let’s give ourselves five blank spaces to work with:
X X X X X
Let’s fill in the first space by picking a value defined by the constraints in the question (1, 2 or 3). Let’s put a 2.
2 X X X X
To be a palindrome, the number has to read the same forwards and backwards. So there would also have to be a 2 in the last space:
2 X X X 2
In the next space let’s also pick a number. Let’s say a 3 this time. Again, for it to be a valid palindrome, we’d also need a 3 in the fourth spot:
2 3 X 3 2
Finally, we have the third spot. Same thing. I’m going for a 2. Does anything have to match this one? Nope. The middle number has no dependents.
And voila. Here is our palindrome: 2 3 2 3 2
Note: even though we must create a 5 digit number,, we are only actually picking 3 of the digits, since the last 2 have to mirror the first 2.
Let’s solve keeping all of that in mind.
How many options do you have for the first number?
It’s not limited at all so you have 1, 2, or 3.
Once you’ve picked the first number, how many options do you have for the second number?
Again: no limit. 1, 2 or 3.
Once you’ve picked the second number how many options do you have for the third?
Free as a bird. 1, 2, or 3.
What about numbers 4 and 5? No options. Those are determined by numbers 1 and 2 (because it’s a palindrome).
3x3x3x1x1 = 27
Correct Answer: E