Lesson: Listing and Counting

Comment on Listing and Counting

Hi Brent,

I am sorry but I am confused with the question "can be created using digits 2 and 7 only". Does that mean that the 3 digits # has to contain both 2 and 7?

Thanks a lot!
gmat-admin's picture

Hi Kate,

For that question, we are basically told that the only digits we can choose from are 2's and 7's. It does not mean that every outcome must contain at least one of each digit.

For example, numbers like 222 and 777 meet the give condition.


Thank you Brent, I felt like I always misinterpret the question and made it harder!! :(


Hi Brent, I am confused with the answer provided for this question?

How many keystrokes are needed to type numbers from 1 to 1000?

A. 2704
B. 2890
C. 2893
D. 3001
E. None of these

My approach:
For 1-9 - 9 strokes
For 10 - 99 - 90*2 strokes
For 100-999 - 999*3 strokes
For 1000 - 4 strokes

Please help me where am I wrong in this?
gmat-admin's picture

The only problem with your solution is the 3rd step. You have 999 integers from 100 to 999, but there are only 900 integers from 100 to 999.

Here's my full solution: https://gmatclub.com/forum/how-many-keystrokes-are-needed-to-type-number...


ari.banerjee's picture

Hi Brent,

Are these questions, listed below the videos, OG questions?

Im planning to do the OG qs in the end so I was curious.
gmat-admin's picture

Good question, Ari.
The source for each question is listed on the right side of the link.
When I say the source is GMAT Club, then either the original poster didn't note the source or it's an original question that was created by GMAT Club staff.
So, in the above links there are no official questions. That said, I often post unofficial questions if they closely resemble official GMAT questions.


ari.banerjee's picture

Thank you.

That makes sense. Mind Boggling qs nonetheless.

For some reason when I solve the OGs I dont encounter such high levels of difficulties. Are OG qs mostly easy with a few 700 level qs inserted here and there?
gmat-admin's picture

I think you're right. Many non-official questions often push the limit difficulty-wise.

Hi Brent,
In below question why cannot there be more numbers sum of whose digits is 3 like say if sum of digits is 12, it still would mean sum is 3?
gmat-admin's picture

Question link: https://gmatclub.com/forum/how-many-integers-are-there-between-2-x-10-5-...

I think you're confusing "a number is divisible by 3" with "the sum of the digits is 3" (although they are, indeed, related)

Take, for example, the number 411033.
Since the sum of its digits is 12 (which is divisible by 3), we know that 411033 is divisible by 3.
HOWEVER, the sum of the digits in 411033 is not 3, so we can't include it in our list of possible outcomes.

Does that help?

Hi Brent,

Wouldn't using the permutation and combination approach be better since by counting and listing numbers, there are chances of missing out on certain combinations?
gmat-admin's picture

Yes, one of the drawbacks to listing and counting is that you might miss an outcome.
However, if you can't find your answer among the answer choices, it's just a matter of rechecking your list to see what outcome(s) you missed.

Also keep in mind that using counting techniques doesn't guarantee you won't make any mistakes either :-)
For example, if you incorrectly use the combination formula to answer a question and can't find your answer among the answer choices, then you might have even more difficulty determining where you went wrong.

In many cases, you'll find that, if you start listing and counting outcomes, you can identify a pattern, which will allow you to stop listing and counting and simply calculate the answer. See the following video for an example of this: https://www.gmatprepnow.com/module/gmat-counting/video/774

So, although the strategy of listing and counting is not without its flaws, it's just another strategy to add to your "mathematical toolbox."
The more tools, the better!

Got that Brent, however to solve questions such as the one mentioned below, would it not help to have an understanding on the fundamentals of permutations and combinations?

Qs: https://gmatclub.com/forum/six-children-arya-betsy-chen-daniel-emily-and-franco-are-to-be-210939.html

If I wanted to learn the PnC approach, could you suggest a module/video or a resource center that I can learn it from?
gmat-admin's picture

Question link: https://gmatclub.com/forum/six-children-arya-betsy-chen-daniel-emily-and...

I'm not suggesting that the strategy of listing and counting is a substitute for counting methods (e.g., combinations, Fundamental Counting Principle, permutations, etc.). However, the strategy is still useful to know, especially in times when you're not sure how to solve the question.

Keep in mind that the above video lesson is just the first of many video lessons on the topic of counting. In subsequent lessons, you'll learn all of the techniques and strategies to solve all counting questions on the GMAT.

By the way, we don't need to know anything about permutations to answer counting questions on the GMAT. In fact, in the following articles, I explain that there are no "true" permutation questions on the GMAT:
- https://www.gmatprepnow.com/articles/combinations-and-non-combinations-%...
- https://www.gmatprepnow.com/articles/combinations-and-non-combinations-%...
- https://www.gmatprepnow.com/articles/does-order-matter-combinations-and-...

I hope that helps.

Hi Brent,


Why is 0000 not a valid solution?
gmat-admin's picture

Question link: https://gmatclub.com/forum/a-palindrome-is-a-number-that-reads-the-same-...

Good question.
When it comes to "n-digit numbers", any zeros that appear before a non-zero digit don't count.
For example 074 is not considered a three-digit number because 074 is the same as 74.
Likewise, since 00555 = 555, 00555 is not considered a 5-digit number; it's considered a three digit number.

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