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Permutation questions are about taking a group of objects and totaling how many ways we can arrange them in specific ways. Here is an example that we will explain later.
In how many ways can a pet shop line up 3 cats and 3 dogs in 6 cages if
the cats must be in the second, fourth, and sixth cages?
I. The Basics: Three Steps to Permutation Clarity
1. Figure out how many places there are to fill.
2. Figure out how many objects potentially can go into each place.
3. Multiply for the answer.
Example
How many outcomes are there when two identical dice are rolled?
Following the steps:
1. Figure out how many places there are to fill
Because there are two dice, there are two places to fill:
__ __
2. Figure out how many objects potentially can go into each place
Because each die has 6 different potential outcomes, we will fill the
spaces accordingly:
_6_ _6_
3. Multiply for the answer
_6_ × _6_ = 36
Example 2
In Country X, three digit area codes are to be given to each town. The first
digit will be any number from 2-9, inclusive, the second digit can only be either
0 or 1, and the third digit can be any number from 0-9, inclusive. How many
different area codes can be issued in Country X?
Following the steps:
1. Figure out how many places there are to fill
Because there are three digits, there are three places to fill: __ __ __
2. Figure out how many objects potentially can go into each place
The question states that the first digit can be any number from 2-9, inclusive.
There are therefore 8 potential options. The second digit can be only 0 or
1, therefore, there are 2 potential options. The third digit can be any number
from 0-9, inclusive, and there are 10 such numbers. The diagram looks like
this:
_8_ _2_ _10_.
3. Multiply for the answer
_8_ × _2_ × _10_ = 160
II. Permutations Without Replacement
Sometimes the number of possibilities decreases instead of remaining the same. With dice, you may role dice as many times as you want, but there will always be 6 possibilities. But sometimes the number of possibilities change in a question.
A student wants to assign 7 different books to 3 spaces, how many different possible
possibilities are there?
Would you calculate _7_ × _7_ × _7_= 343 like above?
How could you if ever time you select a book, the number of possibilities decreases?
Use Logic
Logic tells us that there are 7 choices for the first book. Then 6 choices for the second book and then 5 choices for the third book. Possibilities decrease as items are selected.
To calculate the total possibilities for the three spaces we multiply 7 × 6 × 5 = 210
Permutations Without Replacement Formula
There is a more specific formula for this that essentially does the same thing as the logic above.
In this formula, n stands for the distinct objects which you are choosing from, r stands for the number of spaces which those n objects can fit into, and P stands for Permutation, and is not an arithmetic part of the equation. The exclamation point(!) after each letter represents the factorial of that number.
Factorial(!) means multiplying a number by every positive integer below it down to 1.
5! = 5 × 4 × 3 × 2 × 1 = 120
3! = 3 × 2 × 1 = 6
If you want to fit 7 books into 3 spaces, and want to know the possible permutations, you would assign 7 to n (since the books are the distinct objects which you are choosing from), and assign 3 to r (since it is the number of spaces that n can fit into).
Therefore the formula would read:
that would be written out as:
We can cancel out 4 × 3 × 2 × 1 in both the numerator and denominator, so we
are left with 7 × 6 × 5 which equals 210.
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800score Tip:
It is your choice as a student whether to rely on either the formula or use logic. 800score provides both approaches, but we suggest logic. The GRE isn't interested in your perfect memorization of the permutation formula, the GRE wants you to have a good intuitive sense of how permutations work.
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III. Replacement or Non-Replacement
The GRE will test your ability to distinguish problems with or without replacement. So you should be very good at identifying which one it is.
Replacement
Potential outcomes that are replaced or constant |
Non-Replacement
Potential outcomes that decrease with each selection |
Rolling a set of 6-sided dice
How many possibilities in 10 dice rolls |
Set of 6-sided dice on paint
How many possibilities in 10 dice rolls on wet red paint. If a painted side shows, then the dice must be re-rolled. |
Pulling from a bag of marbles and then putting them back
A student pulls a marble from a bag and then puts it back in the bag. If he does this four times and the bag contains 9 green marbles and 9 blue marbles, how many different possibilities are there? |
Pulling from a bag of marbles
A student pulls 4 marbles from a bag that contains 9 green marbles and 9 blue marbles, how many different possibilities are there?
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Combination of safe or lock
How many possibilities for a combination lock with 40 numbers that requires 3 selections. |
Sum of the combination of a safe where numbers can't be repeated
How many possibilities for a combination lock with 40 numbers that requires 3 selections and cannot have the same number twice. |
Pulling from a repeatedly shuffled deck of cards.
4 cards are pulled from a deck where the dealer shuffles the deck and replaces the card immediately after each card is pulled. |
Pulling from a deck of cards.
4 cards are pulled from a deck of cards. |
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800score Tip:
Keep in mind that the GRE's effectiveness as a test is a function of it's ability not to be "beaten" by standard test preparation.
So it is quite logical for the GRE to set up trick questions specifically to penalize over-preparation. Our test prep gnomes who have recently taken the GRE warn us that questions about pulling cards from a deck or marbles may not be what common "cards" or "marble" questions are usually about.
So if you have done dozens of generic questions with marbles and you assume it must be a permutations without replacement question... be warned: it may not be that way on test day.
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