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In this lesson, you will study how to use combinations and permutations.
Two other useful techniques that aid in counting are permutations and combinations.
In the earlier example, we wanted to find the ways that 7 books could be arranged in 7 spaces on a bookshelf. What if we had only 5 spaces instead of 7 in which to place the 7 books?
We still have 7 choices for the first slot.
There are 6 choices for the second slot.
And so on.
There are 
possibilities.
In this example, the order of the books does matter. If we changed the order, we would change the element of the sample space, which is the set of all possible outcomes. In this situation, we can simplify the computation by using a permutation 
.
The formula for permutations is
This formula describes the number of ways to arrange r elements out of n elements.
There are 36 people in the 6th grade class. How many different ways can the teacher line up 14 students?
The correct answer is D. Instead of using the counting principle directly and writing 14 blanks in which to label possible choices, we can save some paper and employ the permutation formula. Because order matters, we need to use the permutation formula.
.
Notice that if we plan to arrange all the elements of our collection, we are back to the original statement of the fundamental counting principle.
, since 0! = 1.
In our example of the 7 books on a shelf, we initially arranged all 7 books.
Order is very important. For example, you can have all of the numbers correct for a safe combination, but you will not be able to open it unless you have them in the correct order. As we saw earlier, permutations count the elements of a sample space in which the ordering matters, such as students in a line, books on a shelf, or the order in which people finish a race. To count elements in a sample space in which the order does not matter, we use combinations.
The formula for combinations is
This formula describes the number of ways to choose r elements out of n elements.
Two restaurants in town offer vegetarian plates and a variety of vegetables from which to choose. Restaurant A offers 6 different vegetables and customers can choose 2 per plate; restaurant B offers 5 vegetables and lets customers choose 3 per plate. Which statement below is correct?
The correct choice is D. The order doesn’t matter when serving vegetables on a plate, so we should use the combination formula. The number of possible vegetarian plates at restaurant A is:
Similarly, the number of possible vegetarian plates at restaurant B is:
.
Because 
, we must be able to determine which operation to use when counting the total elements of a sample space. This choice hinges on the question of ordering.
Consider a desired result. If changing the order does not change your desired result, then use combinations. If your desired result depends on the order, use permutations.
So if 10 people are running a race and we want to know which group of 3 is the first across the finish line, we would use combinations. If we want to know exactly who came in 1st, 2nd, and 3rd, we would use permutations.
Make a note of the following, it’s worth memorizing!
When order does matter, use permutations 
. Think “n arranging r.”
For groups where order does not matter, use combinations 
. Think “n choosing r.”