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Tutorial of Combinatorics

Today’s topic is permutation. What is permutation you might ask? Well, permutation, as it is defined in any textbook, is an arrangement of a set of distinct symbols in a line in some order, in which the way you order them is important. For example, if you have a set of number {1, 2, 3, 4, 5} and arrange them as follows:

1, 2, 3, 4, 5 is one permutation.

1, 3, 2, 4, 5 is another permutation and so on.

In this kind of combinatory, we also want to know how to order just some of the elements of the set. This ordered arrangement of some elements “r” is called an r-permutation.

Let say we have the set above {1, 2, 3, 4, 5} and we want to choose only 2 of those 5 elements to arrange them in some ordered. We arrange them as 5, 2 or 3, 4 etc. This arrangement is called a 2-permutation of this set of 5 elements. If we choose 3 instead of 2, it would be called a 3-permutation and so forth. “The number of r-permutations of a set with n elements is denoted by P(n, r).” (Discrete Mathematics and Its Applications, Pg. 321). We can find P(n,r) using the product rule and coming up with this theorem

P(n, r) = n(n-1)(n-2)…(n-r+1) = n!/(n-r)!.

This is known to be true because we have n elements in the set and it can be chosen in n different ways. The second we have n-1 ways of chosing it because we took off one element for the first permatutation. The samel goes for the n-2, n-3… elements until there is only n-(r-1) left. Then, multiplying it all together because we are ordering them in certain order. Thus, the theorem holds true.

To illustrate this problem, let look at an example:

Suppose that you are the host for a game show and have 20 contestants on your show. You want to know how many ways can you one person wins the grand prize and another person wins the runner-up prize.

So you have 20 contestants, which are n elements. You have 2 prizes, the grand and runner-up prizes. Since, you want to know who gets the grand prize and who will receive the runner-up prize. So, order is important. Using the above equation, P(20, 2) = 20*19 or 20!/(20-18)! we get 380 ways of chosing 2 people to win the grand prize and the runner-up prize.