Lesson: The Fundamental Counting Principle

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Question at http://gmatclub.com/forum/a-palindromic-number-reads-the-same-forward-and-backward-214196.html is not clear to me. Could you please help me on this?

How many 6-digits number are Palindromic numbers? A Palindromic number reads the same forward and backward, example 12121.
A) 100
B) 610
C) 729
D) 900
E) 1000
gmat-admin's picture

I'm not a huge fan of that question because it first starts by talking about 6-digit numbers and then gives an example of a 5-digit number that works. However, my solution appears on that same page.
The basic idea is that, once we have selected the first 3 digits of the number, the last 3 digits automatically follow.
For example, if the first 3 digits are 356---, then (to be a palindrome), the last 3 digits must be ---652, so we get the number 356652.
Likewise, if the first 3 digits are 197---, then (to be a palindrome), the last 3 digits must be ---791, so we get the number 197791.

Hey Brent! I love your videos so much. They are of great help to me!
My question is, how commonly is this topic tested on the GMAT? Is this a frequently tested topic? Is it advisable to skip this one topic out of all others or will it hurt my score significantly?

gmat-admin's picture

Glad you like the videos!

Counting questions aren't that common on the GMAT.

The number of questions YOU see on test day will depend on how well you're doing on the quantitative section. If you're doing really well, you MIGHT see 2 or 3 counting questions. If you're not doing well, you'll see FEWER counting questions.

If your prep time is limited, I suggest that you master the The Fundamental Counting Principle strategy, since it can be used for the majority of counting questions on the GMAT.

In this question below:
Team A and Team B are competing against each other in a game of tug-of-war. Team A, consisting of 3 males and 3 females, decides to lineup male, female, male, female, male, female. The lineup that Team A chooses will be one of how many different possible lineups?
-Should I assume that 'decides to line up..." means that it is mandatory that the lineup be as such?
-I interpreted decided lineup in a way that the answer is 6! but it looks like there is restrictions here (answer being 3*3*2*2*1*1) but the language doesn't seem that clear. Thoughts? Is this an official GMAT lingo?
gmat-admin's picture

6! represents the number of ways to arrange 6 people WITHOUT any restrictions (6 ways to fill the first position, 5 ways to fill the second position , etc).

However, the restriction says that the first position must be a male. So, there are only 3 ways to fill that first position. Etc.

The question could be worded MUCH better.

Dear Brent,

I was trying to solve questions similar to:
The Carson family will purchase three used cars. There are two models of cars available, Model A and Model B, each of which is available in four colors: blue, black, red, and green. How many different combinations of three cars can the Carsons select if all the cars are to be different colors?

I am simply not able to wrap my head around any of approaches being used to solve this question. Neither the stages method explained in the GMATPrep videos nor the Manhattan slot method.

When I tried solving the it using the stages method, i went by solving as described below:

Stage 1: Car1 - 4 colors x 2 models
Car 2 - 3 colors x 2 models
Car 3 - 2 Colors x 2 models.

I am having a hard time understanding the repetition part of these kind of questions and always get stuck.
gmat-admin's picture

Check out my solution below, and please let me know if it makes sense: http://www.beatthegmat.com/combinatorics-problem-t267079.html

for the above problem -
cars and models ad colors.
The leaf diagram gives me 24.
3 cars * 2 models * 4 colors = 24.
What am I missing ?

gmat-admin's picture

First off, your solution does not take into account that the 3 cars must be 3 different colors.

To help better understand why your approach, please break your solution into steps/stages and tell me what you are doing at at point in the solution.

ASIDE: After answering a few questions involving the Fundamental Counting Principle, some students fall into the habit of simply finding the product of all of the numbers that appear in the question. This is not a good strategy.


I did not understand your answer, could you help me with this below question?

Thanks and regards
gmat-admin's picture

Hi Brent,
for this question: How many possible ways can 3 girls (Rebecca, Kate, Ashley) go on a date with 3 boys (Peter, Kyle, Sam)?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Assuming a date requires 1 boy + 1 girl
Stage 1 : selecting a boy. Can be done in 3 ways
Stage 2 : selecting a girl. Can be done in 3 ways

My answer is 9. However thats not one of the options. What gives ?
gmat-admin's picture

Question link: https://gmatclub.com/forum/how-many-possible-ways-can-3-girls-rebecca-ka...

The problem with your approach is that you are answering a different question.

You are answering the question "In how many ways can ONE girl (from Rebecca, Kate, Ashley) go on a date with ONE boy (from Peter, Kyle, Sam)?"

If that were the question, then the correct answer would, indeed, be 9.

However, in the original question, all 6 people are going on the date, and each person from one group gets paired with someone from the other group.

Does that help?


yes thanks a ton !

Hello for this question I completed this way and still got the same answer as you can you please confirm that my solution is right

A B C D E Number choices 1,2,3,4,5,6
(c is the middle digit )
A = 6 options
B = 5 options
C = 1 option ( 3 odd numbers in total if A was odd then B would have 2 options leaving C with 1 option)
D = 4
E = 3

6*5*1*4*3 = 360

Is my solution correct also?

gmat-admin's picture

I can help as soon as you tell me which question you're referring to :-)



thought I could paste the question link here. The question is:

How many different five-digit codes can be picked from the digits 1 through 6 if the middle digit must be odd and no two digits might be the same?

A) 420
B) 360
C) 180
D) 120
E) 60
gmat-admin's picture

Link: https://www.beatthegmat.com/picking-a-5-digit-code-with-an-odd-middle-di...

When you use the Fundamental Counting Principle, you must be sure to deal with the most restrictive rule first. In this question, the most restrictive rule says that the middle digit must be odd.
If you don't deal with that first, then you end up with scenarios like you have where you say "C = 1 option ( 3 odd numbers in total if A was odd then B would have 2 options leaving C with 1 option)"

While you did, indeed, reach the same answer, I'm not entirely sure how you dealt with this mathematically.
I would avoid that approach in the future. Instead, be sure to deal with the most restrictive rule first.


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