# 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

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
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
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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