List all the permutations of {a, b, c}

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Like in other exercises, we should start with the definitions.

“A permutation of a set of distinct objects is an ordered arrangement of these objects.”

It is also important to know beforehand how many of these ordered arrangements are there, so we don’t forget anyone when asked this type of question.

Theorem: “If n is a positive integer and r is an integer with 1 ≤ r ≤ n, then there are P (n, r ) = n(n − 1)(n − 2) · · · (n − r + 1) r-permutations of a set with n distinct elements.”

Corollary: “If n and r are integers with 0 ≤ r ≤ n, then P(n,r) = n! / (n−r)!”

The previous definition, theorem, and corollary are from the textbook Discrete Mathematics and its Applications by Rosen.

Now we have the tools that we need to answer the exercise.

As a first step, let’s find out how many permutations we need to write.

In this case, we have to write all the 3-permutations of 3 elements. Notice that the set {a,b,c} has 3 elements.

According to the formula, P(3,3) = 3!/(3-3)!=6/1=6.

Now we know that we must write a total of 6 permutations.

Also, from the first definition, we know that a permutation is an ordered arrangement of objects, so the order matters. This means that “ab” is a different permutation than “ba”.

Considering the previous reasoning, the permutations are:

  1. abc
  2. acb
  3. bac
  4. bca
  5. cab
  6. cba

Another approach to solve this type of exercise is by noticing that we can also use the multiplication rule to find out the total number of permutations.

If we change the question to “In how many ways can we arrange the elements abc?”.

You now know, because you solved similar exercises, that you can choose the first element in 3 ways, the second element in 2 ways, and the third element in 1 way.

Because for every 3 ways you can choose the first element, you have 2 ways of choosing the second one, and 1 way of choosing the third one, we realize that we can apply the multiplication rule.

Therefore, we have 3x2x1=6 possible ways of arranging the three elements. From there, you know that you won’t miss any arrangements.

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