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You have 4 math books and 5 history books to put on a shelf that has 5 slots. In how many ways can the books be shelved if the first three slots are filled with math books and the next two slots are filled with history books?

We first do the problem using the multiplication axiom.

Since the math books go in the first three slots, there are 4 choices for the first slot, 3 for the second and 2 for the third. The fourth slot requires a history book, and has five choices. Once that choice is made, there are 4 history books left, and therefore, 4 choices for the last slot. The choices are shown below.

4 3 2 5 4

Therefore, the number of permutations are 4 3 2 5 4 = 480 .

Alternately, we can see that 4 3 2 is really same as 4 P 3 , and 5 4 is 5 P 2 .

So the answer can be written as ( P 3 ) ( 5 P 2 ) = 480 .

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Clearly, this makes sense. For every permutation of three math books placed in the first three slots, there are 5 P 2 permutations of history books that can be placed in the last two slots. Hence the multiplication axiom applies, and we have the answer ( 4 P 3 ) ( 5 P 2 ) .

We summarize.

  1. Permutations

    A permutation of a set of elements is an ordered arrangement where each element is used once.

  2. Factorial

    n ! = n n 1 n 2 n 3 3 2 1 size 12{n!=n left (n - 1 right ) left (n - 2 right ) left (n - 3 right ) dotsaxis 3 cdot 2 cdot 1} {} .

    Where n size 12{n} {} is a natural number.

    0 ! = 1 size 12{0!=1} {}
  3. Permutations of n size 12{n} {} Objects Taken r size 12{r} {} at a Time

    n Pr = n n 1 n 2 n 3 n r + 1 size 12{n"Pr"=n left (n - 1 right ) left (n - 2 right ) left (n - 3 right ) dotsaxis left (n - r+1 right )} {} , or n Pr = n ! n r ! size 12{n"Pr"= { {n!} over { left (n - r right )!} } } {}

    Where n size 12{n} {} and r size 12{r} {} are natural numbers.

Circular permutations and permutations with similar elements

Section overview

In this section we will address the following two problems.

  1. In how many different ways can five people be seated in a circle?
  2. In how many different ways can the letters of the word MISSISSIPPI be arranged?

The first problem comes under the category of Circular Permutations, and the second under Permutations with Similar Elements.

Circular permutations

Suppose we have three people named A size 12{A} {} , B size 12{B} {} , and C size 12{C} {} . We have already determined that they can be seated in a straight line in 3 ! size 12{3!} {} or 6 ways. Our next problem is to see how many ways these people can be seated in a circle. We draw a diagram.

This figure shows how the people named A, B, and C can sit in a circle in different ways.

It happens that there are only two ways we can seat three people in a circle. This kind of permutation is called a circular permutation. In such cases, no matter where the first person sits, the permutation is not affected. Each person can shift as many places as they like, and the permutation will not be changed. Imagine the people on a merry-go-round; the rotation of the permutation does not generate a new permutation. So in circular permutations, the first person is considered a place holder, and where he sits does not matter.

Circular permutations

The number of permutations of n size 12{n} {} elements in a circle is n 1 ! size 12{ left (n - 1 right )!} {}

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In how many different ways can five people be seated at a circular table?

We have already determined that the first person is just a place holder. Therefore, there is only one choice for the first spot. We have

1 4 3 2 1

So the answer is 24.

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In how many ways can four couples be seated at a round table if the men and women want to sit alternately?

We again emphasize that the first person can sit anywhere without affecting the permutation.

So there is only one choice for the first spot. Suppose a man sat down first. The chair next to it must belong to a woman, and there are 4 choices. The next chair belongs to a man, so there are three choices and so on. We list the choices below.

1 4 3 3 2 2 1 1

So the answer is 144.

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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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