## When to Use Each Counting Rule

[Music]

all right we are going to look at when

do we use each of the counting rules we

only have three so this is a little bit

simpler but this will kind of look

through the questions you should kind of

problems that use counting rules okay it

will review our counting rules as we do

this as well the first question you

should ask yourself on a counting roll

problem is is there repetition is

repetition allowed there's repetition

allowed so are you able to use something

again once you select it can you select

it again so for example and how many zip

codes are possible well you're able to

use the number one every time you can

have a zip code of five one so that

would be yes repetition is allowed and

when repetition is allowed we have to

use this general multiplication rule for

counting

okay the permutation role and the

combination role do not allow for

repetitions they use them up as you go

through and so if you do allow

repetition then you have to use the

multiplication rule and the

multiplication rule says that you take

the the number of times that the first

event can happen times the times the

number of events that the second event

can happen and so on and in our sequence

of events and that will tell us the

number of ways that we can have our

sequence of events okay let's look at an

example of the zipcode problem that I

mentioned if we have a zip code which is

five digits how many possible zip codes

are there

okay in zip code I do allow for

repetition my zip code is three seven

three one one so it uses the seven twice

that uses the one twice so we do have

some repetition that is allowed there

and so we all have to use a

multiplication rule and so we take the

we look at each of it and each event

here would be choosing each number so

how many ways can we choose the first

number how many ways can we choose the

second number those are our events and

then we multiply the number of ways that

each event can happen together so how

many ways can that first event can

happen how many ways can we choose the

first number of a zip code well there

are ten digits

don't forget about zero so there are ten

single digits that you can use that

could be that first digit we'll keep it

simple and we will allow zero to be one

of them usually not okay and then how

many ways can we choose that second

digit well there's still ten digits how

many ways can we choose the third we can

use the same digits again so there's ten

ways there forth and set so this is the

number of ways that we can choose a zip

code number of different possible zip

codes that we have okay 100 thousand

possible zip codes and we could even

make this a nine or we can so that we

don't have a zero at the beginning and

different things like that but just

simply looking at a multiplication rule

problem like this you can move on with

it as well okay but if the answer is no

we cannot use repetition then we might

be able to use some permutations or

combinations but which one so the next

question you need to ask is this order

matter

we know that that was permutations that

we do count the orders we owe do count

all different orders of the same same

set of objects so we count all the

different arrangements so order matters

with a permutation but it doesn't matter

with a combination so what we're really

asking yourself here is do we want to

count all the different arrangements of

the same group of objects or people or

whatever all might that as well do you

want to count all different arrangements

do we want to count all different

arrangements of the same group

if we do if order does matter the answer

to this question is yes permutations do

that and the thing is search of this

question is no I don't I don't care

about all the different arrangements and

combinations will do that okay

the permutation we call a lot of times

in class we call it NPR okay we have a P

in there and this you can do in your

calculator the formula for that you can

just enter that in and then combination

is in CR okay so that's what you are

example of both of these when order

matters and when order does not matter

so let's say that we want to pick three

students from a class of ten but we are

going to assign them president vice

president and secretary okay so let's

pick three students from a class of ten

we're going to assign them different

roles so in that case if we want to

count how many different ways we can

select these three students we do care

about the order I mean it makes a

difference whether I'm president or

whether I'm secretary okay

even if I'm picked if I'm picked it

matters which one I am husband or

secretary so I do care about order I

want to count selecting the same group

of three students but arranging them

differently into these three different

roles so that's another thing you want

to kind of look at when you look at a

permutation that clue you into

permutation is are there different roles

assigned or are they winning something

different then you're probably wanting

to count the different arrangement okay

if I said there are ten students we're

selecting three of them so this would be

a ten P three okay like ten students

choose the president vice president

secretary how many different

groups can we come up with that would be

a 10 p3 there's 5 different roles once

against those permutations and this

sends that being 720 ways okay what kind

of example of a combination role where

we don't care about the order if we were

just taking these 10 students and we

were picking 3 of them to form a

assigning them different roles we're

just picking them it doesn't it doesn't

matter what number I'm picked okay it

just matters if I'm picked or if I'm not

so I don't care about the order in this

case because I'm just being on a three

person committee that doesn't have any

roles that we're all doing the same

thing so in that case we know that we're

going to be using a combination the

order doesn't matter of course

repetition is not allowed so we're over

here and that would be 10 C 3 so how

many ways can we choose three people

from 10 we don't like how all the

different arrangements of the same three

people because they're all doing the

same thing so this one isn't being 120

much smaller because we're not counting

all the different arrangements of those

same three people so this one should

always be smaller than this one if you

are in and your are are the same so it's

another check as to which one you should

be using if you if you get confused as

to which link ounce the order in which

does it keep the same n and the same R

and your permutation is going to be

bigger because it does count all the

different arrangements another thing I

want to note is that anytime you're

using a permutation you could also use

the multiplication role

the permutation just keeps things a

little bit simpler if we were to have

large numbers like if we were to have 50

students and we're choosing 10 of them

it's a little bit of a pain to use the

multiplication rule and when we could

just simply put into our calculator 50p

10 so we could use the multiplication

rule here even and take 10 times 9 times

8 because there's 10 ways to choose a

smart student repetition is not allowed

because we couldn't choose the same

student again so you have 9 here and

then eight is also results in 720 okay