# 4.6 Counting rules - university of calgary - base content - v2015  (Page 2/6)

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$4\times 3\times 2\times 1=24$

This can be represented by 4! (read “four factorial”).

$4!=4\times 3\times 2\times 1=24$

## Factorials

The exclamation symbol after a natural number indicates to multiple a series of descending natural numbers from n to 1 .

$n!=n\times n-1\times n-2\times ...\times 1$
0!=1

Suppose that we have five members on a committee.

1. If there are two positions on the committee of president and vice president, how many different ways could the positions be filled?
2. If there are five positions on the committee, how many different ways could you fill the five positions?

1. There are 20 different ways to fill the position of president and vice president. There are five members to choose from for the first position and if we choose a member to be president, we now have four members to choose from for the next position.

$5\times 4=20$

2. $5!$ = $5\times 4\times 3\times 2\times 1=120$

## Using ti-83,83+,84,84+ calculator

• Enter 5.
• Press MATH.
• Arrow across to PRB.
• Press ENTER.
• Arrow down the list to 4:!
• Press ENTER.

20 students have volunteered to be class representatives (reps).

1. If the teacher randomly selects two students to be class representatives, how many groups of two can be formed?
2. If the teacher randomly selects four students to be class representatives, how many groups of four can be formed?
3. If the teacher selects all 20 students to be class representatives and each one is assigned to a group of students, how many different ways can theclass representatives be assigned?

1. $20\times 19=380$
2. $20\times 19\times 18\times 17=116280$
3. $20!$

Let’s look again at the example of hanging pictures along a wall. Suppose you decide that you are only going to hang two of the pictures in a row. We saw that if we choose one picture to hang first, we are now left with three choices of pictures for the next position on the wall. Using the multiplicative rule, there are 43 = 12 possible arrangements of pictures for the first two positions on the wall.

There are four different pictures and we are selecting only two and arranging them in a specific order. This is known as a permutation .

## Permutation

Permutation is the arrangements of r elements in a different order chosen from n distinct available items. Below are different ways that a permutation can be represented.Insert paragraph text here.

${P}_{r}^{n}={}_{n}{P}_{r}=\frac{n!}{(n-r)!}=n(n-1)n(n-2)\mathrm{...}(n-r+1)$

For the picture example, there are four pictures ( n = 4) and we are selecting two pictures ( r = 2) and arranging them in a specific order. Using the multiplication rule, we have seen that the answer is 12. Using permutation, we can see that we get the same result.

${P}_{2}^{4}=$ ${}_{4}{P}_{2}=$ $\frac{4!}{(4-2)!}=$ $\frac{4!}{2!}=$ $\frac{4\times 3\times 2\times 1}{2\times 1}=$ $4\times 3=12$

There are 4! different ways of arranging the four pictures on the wall. We divide by the number of ways of arranging the items that are not selected because we only care about the arrangement of the items selected.

There are 4! different ways of arranging the four pictures on the wall. We divide by the number of ways of arranging the items that are not selected because we only care about the arrangement of the items selected.

For example, let’s label the pictures A , B , C and D . If we write out the sample space for arranging 4 pictures along a wall, we get the sample space

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