# 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

Use [link] with these instructions.

• 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

#### Questions & Answers

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x+2y-z=7
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An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? Kala Reply lim x to infinity e^1-e^-1/log(1+x) given eccentricity and a point find the equiation Moses Reply 12, 17, 22.... 25th term Alexandra Reply 12, 17, 22.... 25th term Akash College algebra is really hard? Shirleen Reply Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table. Carole I'm 13 and I understand it great AJ I am 1 year old but I can do it! 1+1=2 proof very hard for me though. Atone hi Adu Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily. Vedant find the 15th term of the geometric sequince whose first is 18 and last term of 387 Jerwin Reply I know this work salma The given of f(x=x-2. then what is the value of this f(3) 5f(x+1) virgelyn Reply hmm well what is the answer Abhi If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10 Augustine how do they get the third part x = (32)5/4 kinnecy Reply make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be AJ how Sheref can someone help me with some logarithmic and exponential equations. Jeffrey Reply sure. what is your question? ninjadapaul 20/(×-6^2) Salomon okay, so you have 6 raised to the power of 2. what is that part of your answer ninjadapaul I don't understand what the A with approx sign and the boxed x mean ninjadapaul it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared Salomon I'm not sure why it wrote it the other way Salomon I got X =-6 Salomon ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6 ninjadapaul oops. ignore that. ninjadapaul so you not have an equal sign anywhere in the original equation? ninjadapaul hmm Abhi is it a question of log Abhi 🤔. Abhi I rally confuse this number And equations too I need exactly help salma But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends salma Commplementary angles Idrissa Reply hello Sherica im all ears I need to learn Sherica right! what he said ⤴⤴⤴ Tamia hii Uday hi salma hi Ayuba Hello opoku hi Ali greetings from Iran Ali salut. from Algeria Bach hi Nharnhar A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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Source:  OpenStax, Introduction to statistics i - stat 213 - university of calgary - ver2015revb. OpenStax CNX. Oct 21, 2015 Download for free at http://legacy.cnx.org/content/col11874/1.3
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