The Fundamental Counting Principle states that if an event has x possible outcomes and another independent event has y possible outcomes, then there are xy possible ways the two events could occur together.
If you are aiming for a score of 700+ on the GMAT, you’ll probably see some harder probability questions involving counting, permutations and combinations. Let’s take a look at some examples:
(Note: If you want some practice, try solving the example questions first before reading the answer.)
Example 1 – Fundamental Counting Principle
1. How many three-digit integers have either 3 or 4 as their tens digit and 6 as their units digit?
To solve this problem, we need to find the possible outcomes for each digit (hundreds, tens, and units) and multiple them. Each digit was 10 possible numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9). The hundreds digit can be any of them except 0 (since a three-digit number cannot begin with 0). The tens digit has only 2 possibilities as stated in the question. The units digit has only 1 possibility. According to the Fundamental Counting Principle, the total number of possible numbers is 9 x 2 x 1 = 18.
Example 2 – Permutations
Permutations are sequences. In a sequence, order is important.
2. How many different ways can six people stand in line?
For the first spot in line, we have six people to choose from, and as we go down the line trying to fill each spot, the number of people we have to choose from will decrease by 1. Therefore we have 6 x 5 x 4 x 3 x 2 x 1 = 720 ways
Example 3 – Permutations
Some harder permutations problems will require you to use this formula:
n = the total number of options
r = the number of options chosen
3. At Martin Luther King High School’s track competition, 14 athletes are competing in the pole vault finals. How many possible options are there for the first 4 finishers?
Here n = 14 and r = 4. Since the order in which the athletes finish matters, we know to use the Permutation formula:
n! / (n – r)! = 14! / (14 – 4)! = 14! / 10! = 14 x 13 x 12 x 11 = 24, 024 options (10! cancels out from the numerator and denominator)
Example 4 – Combinations
Combinations are groups. Order doesn’t matter. The Combination formula is only slightly different from the Permutation formula:
n = total number of options
r = the number of options chosen
4. Lisa took 14 photos with her new digital camera. She wants to choose 10 of them to put on her Facebook profile. How many different groups of photos are possible?
Since the question asks about groups and not the order of the photos, we know this is a Combination problem.
n! / r! (n – r)! = 14! / 10! (14 – 10)! = 14! / 10! 4!
= 14 x 13 x 12 x 11 / 4 x 3 x 2 x 1 = 1,001 different groups (Again, 10! cancels out)
Always remember to ask yourself whether order matters to the problem you are trying to solve, and don’t forget the Fundamental Counting Principle! The GMAT may also combine one or more of these concepts into the same problem. Check out some games on Grockit for more examples.
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