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Combinations and Permutations

How Can We Tell the Difference Between Them?

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To figure out some probability problems, we must be able to count. Suppose we are given a total of n distinct objects and want to select r of them. This touches directly on an area of mathematics known as combinatorics, which is the study of counting. Two of the main ways to count are called permutations and combinations. These concepts are closely related to one another and easily confused.

How can we tell the difference between a combination and permutation? The key idea is that of order. A permutation pays attention to the order that we select our objects. The same set of objects, but taken in a different order will give us different permutations. With a combination we still select r objects from a total of n, but the order is no longer considered.

An Example of Permutations

To distinguish between these ideas, we will consider the following example: how many permutations are there of two letters from the set {a,b,c}?

Here we list all pairs of elements from the given set, all the while paying attention to the order. There are a total of six permutations. The list of all of these are: ab, ba, bc, cb, ac, ca. Note that as permutations ab and ba are different because in one case a was chosen first, and in the other a was chosen second.

An Example of Combinations

Now we will answer the following question: how many combinations are there of two letters from the set {a,b,c}?

Since we are dealing with combinations, we no longer care about the order. We can solve this problem by looking back at the permutations, and then eliminating those that include the same letters. As combinations, ab and ba are regarded as the same. Thus there are only three combinations: ab, ac, bc.

Formulas

For situations we encounter with larger sets it is too time consuming to list out all of the possible permutations or combinations and count the end result. Fortunately there are formulas that give us the number of permutations or combinations of n objects taken r at a time.

In these formulas we use the shorthand notation of n!, called n factorial. The factorial simply says to multiply all positive whole numbers less than or equal to n together. So, for instance, 4! = 4 x 3 x 2 x 1 = 24. By definition 0! = 1.

The number of permutations of n objects taken r at a time is given by the formula:

P(n,r) = n!/(n - r)!

The number of combinations of n objects taken r at a time is given by the formula:

C(n,r) = n!/[r!(n - r)!]

Formulas at Work

To see the formulas at work, let’s look at the initial example. The number of permutations of a set of three objects taken two at a time is given by P(3,2) = 3!/(3 - 2)! = 6/1 = 6. This matches exactly what we obtained by listing all of the permutations.

The number of combinations of a set of three objects taken two at a time is given by:

C(3,2) = 3!/[2!(3-2)!] = 6/2 = 3. Again, this lines up exactly with what we saw before.

The formulas definitely save time when we are asked to find the number of permutations of a larger set. For instance, how many permutations are there of a set of ten objects taken three at a time. It would take awhile to list all the permutations, but with the formulas we see that there would be:

P(10,3) = 10!/(10-3)! = 10!/7! = 10 x 9 x 8 = 720 permutations.

The Main Idea

What is the difference between permutations and combinations? The bottom line is that in counting situations that involve an order, permutations should be used. If the order is not important, then combinations should be utilized.

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