GMAT Math: Combinations and Permutations
Learn to differentiate combinations and permutations
English can often be quite confusing. It uses the same words for slightly different scenarios. For example, we may have a salad that is a combination of lettuce, tomatoes and carrots. We are not concerned which order the ingredients are in, it may be "carrots, tomatoes and lettuce" or "tomatoes, carrots and lettuce", it's still the same salad.
In a different scenario we may have a combination to a lock. If the combination is "24, 45 and 32", then here the order is very important. Changing the order of numbers will not open the lock; we must stick to the original arrangement.
On the GMAT, we need to differentiate between combinations that must be in a certain order (like the lock) and sequences where the order is not important (salad example). We use the word "Combination" for cases where the order does not matter and we use the word "Permutation" for situations where it does matter.
To help you remember you can use this phase:
A Permutation is an ordered Combination
A Permutation, as we already know, is a possible order in which we put a set of objects. Let's say we have a bookshelf upon which we want to arrange 5 books. How many different arrangements can I make using the books? Or thinking about it slightly the GMAT way: how many permutations are there?
There is a simple formula that we can use here; however, let's see if we can work it out the logical way first.
There are 5 books. Let's call these books A, B, C, D and E. Each one of these books can be in the first position, so that gives us 5 options for place one on the shelf.
That leaves us with 4 books that could possibly go into position two.
So the total number of arrange for the first two positions would be:
5×4 = 20 different permutations.
If were to write this out, it would be: AB, AC, AD, AE, BA, BC, BD, BE, CA, CB, CD, CE, DA, DB, DC, DE, EA, EB, EC, ED
If there is a book in position one and a book in position 2, then that would leave us with three books to choose from for the third position, 2 book to choose from for the fourth position and only one option for the final position.
Given that there are 5×4 permutations for the first two positions, there will be 5×4×3 for the first three positions, 5×4×3×2 for the first four positions and 5×4×3×2×1 permutations for all 5 positions on the shelf.
So the total number of different arrangements for 5 books, or permutations is going to be:
Or as we write it in math: 5!
The exclamation mark does not mean we are very excited about number 5; it the factorial, which basically means a product of all integers from 1 up to 5.
5! = 5×4×3×2×1
The simple formula for the number of permutations for n objects is thus:
Let's take another example. There are 6 people in a line. In how many different ways can the line be formed?
Again this is a simple permutations question, where n will be the number of people in a line: n = 6. The answer will be: 6! which is equal to: 6×5×4×3×2×1 = 720
Ok, that's it for the basics. GMAT permutations are somewhat harder than simply arranging 6 people. A typical GMAT permutation problem will involve calculating the number of permutations within a larger set of numbers. An example of this may be:
Out of a set of 9 books, 5 are to be stacked onto a shelf. How many different arrangements are there to stack 5 books?
Again this question is asking us to calculate permutations. It is complicated however, by some extra books, which we did not have in the previous example. There is another formula for this; however, once again we will work out the logic of the formula first. In this example, there are 9 possible books that can go in position one. That would leave 8 possible choices for position two, 7 books to choose from for position three, 6 books for position four and finally 5 books for the last position.
So the total number of permutations will be 9×8×7×6×5 = 15,120
This can also be calculated using a more universal permutations formula:
Number of permutations =
Where n, is the number of items in the set and r, is the number of elements to be selected.
In this example, there are 9 books so n = 9. Likewise we have to arrange the books into groups of 5, so r = 5.
Using the formula we get:
Number of permutations =
That is the same calculation we performed above, and thus would give us the same answer.
In practice it's much quicker to use the formula in most cases; however, it is important to understand why the formula works. Memorise the formulas for both simple and more complex permutations.
Now that we understand permutations, combinations will be straightforward. Remember, in combinations the order does not matter; 1, 2, 3 and 2, 3, 1 are considered the same, since they combine the same numbers.
Let's look at a similar example with slightly different wording:
Out of a set of 9 books, 5 are to be stacked onto a shelf. In how many ways we can choose 5 books out of 9?
This question makes it obvious we are looking for combinations and not permutations, so we know order does not matter.
We can therefore use the equation for Combinations:
Where n, is still the number of items in the set and r, is still the number of elements to be selected.
Notice it is very similar to the equation for permutations except for one small change in the denominator. We have added an extra r! This will allow us to eliminate all the arrangements that are the same combination; 1, 2, 3 in combinations is the same as 2, 3, 1.
That will leave us with all the different combinations without including the various permutations within each combination.
In this question we have 9 books from which we must arrange 5. The number of books will therefore be n: n = 9. The number of books to be selected will be r: r = 5
After inserting these numbers into the combinations equation we are left with:
Number of Combinations =
Note that the number of combinations will always be lower than the number of permutations, when we are looking at larger sets.
Remember the phrase; A Permutation is an ordered Combination.
Permutation questions require us to think about the order of a sequence whereas combination questions do not. We first have to figure out whether the question is asking for a combination or permutation and then use the right equation.
You will naturally have to learn the formulas mentioned in this post; however, it is equally important to understand why those equations work. GMAT combinations and permutations questions can be complicated, but being able to understand what the formula actually calculates, will help immensely in making those final adjustments to get the right answer.
Practice also makes perfect, so practice, practice, and practice!!!
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- GMAT Math: Basic Probability
- GMAT Math: Comparing Fractions
- GMAT Math:Picking Numbers
- GMAT Math: Backsolving