Combinations + Permutations

[halcyon ruin]

Can’t really understand this topic, struggling to find resources online too

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[halcyon ruin]

I’m assuming that combinations would be used in this case

[deft wedge]

Okay. Do you know how combinations work?

[halcyon ruin]

Yeah it’s the amount of different ways something can be chosen

[deft wedge]

Good. So what do you think you should do here, as a strategy?

(Note that we want to choose at least one of each game.)

[halcyon ruin]

Need atleast one type so u can have either 1 word game, 1 quiz and the rest saga or other way round for all three

[halcyon ruin]

But not sure how to put this into a formula where all situations are considered

[deft wedge]

Okay.

You know about the permutation and combination functions?

[halcyon ruin]

Yeah

Well kind of

[deft wedge]

Okay.

Well, to be concise, it basically chooses the amount of something (k) out of the possible amount you have (n).

Permutations have order matter, and combinations don't.

[halcyon ruin]

Yeah I understand that part but how would you order it in a way where there’s atleast one of each type

[deft wedge]

You combine possibilites using addition.

[halcyon ruin]

So you’d combine the possibility of every single combination there is?

[deft wedge]

Not necessarily.

Note that we only need 5 apps.

So, we already have 3 spots taken up.

[halcyon ruin]

Ohh

[deft wedge]

Thus, 2 spots remain. Correct?

[halcyon ruin]

Yeah

[deft wedge]

You think you can finish this one on your own?

[halcyon ruin]

No

[deft wedge]

There's really only 2 cases here.

[halcyon ruin]

2 spots left

[deft wedge]

That's alright, then.

[halcyon ruin]

Can either have one of one type and one of another

Or two of the same type

[deft wedge]

And what function should we use? Permutation or combination?

[halcyon ruin]

Combination

[deft wedge]

Good.

[halcyon ruin]

So you’d have 4c2 possibilities for one type?

• 4c1

[deft wedge]

For the quizzes, yes.

[deft wedge]

Explain why.

[halcyon ruin]

One quiz is already been used where one of each was required

So 4 left to choose from

And you can have 2 from the 4 or 1 from the remaining 4

[halcyon ruin]

But would you times it or add it?

[deft wedge]

Well, let's split this problem into cases first, to not confuse ourselves.

2 cases:

1. 2 apps of the same type
2. 1 app of each type, which has 3 possible combinations.

So, can you work out the possibilities for the first case?

[halcyon ruin]

15c2 + 4c2 + 1 ?

Which is 112

[deft wedge]

Yep.

Okay, here is a hint for the second; when dealing with the pairs themselves, multiply the possibilities.

[halcyon ruin]

So 15c1 * 4c1 ?

[deft wedge]

Yep.

Then?

[halcyon ruin]

• 4c1 * 1c1

• 1c1 * 15c1

[halcyon ruin]

Don’t really understand that part

[deft wedge]

Which?

[halcyon ruin]

How 1 app of each type means 15c1 * 4c1 + 4c1 * 1c1 + 1c1 * 15c1

[deft wedge]

Hmm. Well, you have different types of phone apps (supposedely), so one isn't necessarily the same as the other. Thus, you need to multiply the possibilities.

[halcyon ruin]

Ok so is this part correct?

[deft wedge]

Since we have 4 apps for quizzes, and 15 for saga games, there are (supposedly) different types of each. So (denoting the n-th quiz game to be Q_n, and the n-th saga game to be S_n), Q_1 and S_1, Q_1 and S_2...Q_2 and S_1...Q_4 and S_15 are all different pairings. This is equivalent to 4*15=60, as we calculated earlier.

[deft wedge]

Yes.

[halcyon ruin]

So there are 60 different ways of having one saga and one quizz app right

[deft wedge]

Yep.

So now, add all those possibilites together.

[deft wedge]

These.

[deft wedge]

This, and the other 2.

[halcyon ruin]

Didn’t work out the possibility of having one of each

I think

[halcyon ruin]

This part

191 possibilities of having either 2 types of the same type and 1 type of 2 different types ?

[deft wedge]

Hmm.

[deft wedge]

,w 15C2+4C2+1

...

[halcyon ruin]

Lol

[halcyon ruin]

That’s 112

[deft wedge]

Yeah.

[deft wedge]

Yes, according to my calculations.

[gray stone]

Oh, isn't this the type of question for which stars and bars are useful?

[halcyon ruin]

So now just need to find out how many possibilities of there being 1 type of each

[halcyon ruin]

Stars and bars?

Explain please it might help me a lot

[deft wedge]

What do you mean?

[halcyon ruin]

Sound useful

[halcyon ruin]

There needs to be atleast one of each type of game

Their installing 5 apps

[deft wedge]

Yes.

[gray stone]

Yeah, a combinatorial method for solving equations of the form x(1) + ... + x(k) = n, where all x-s and n are natural numbers. Two main cases are all x-s being positive (so, your case, pretty much) and all of them being nonnegative.
https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

[deft wedge]

Aha. So you add that as well.

[halcyon ruin]

Yeah but how would you work that out

[deft wedge]

(2C1+5C1+16C1)+191

Do you see why?

[halcyon ruin]

Yeah 16 options and you want one of them

Why add tho and not *

[deft wedge]

Ah, you're right. Sorry.

[halcyon ruin]

Wdym by x-s mean and what is n?

[deft wedge]

Yeah, my bad.

[gray stone]

n is the right hand side of the equation, and x(1), ..., x(k) are variables.

[halcyon ruin]

Is it supposed to be (2C1 * 5C1 * 16C1) + 191?

[deft wedge]

Yes.

[halcyon ruin]

I don’t really understand it from the Wikipedia link, is there a simple way of putting the stars and bars method?

[halcyon ruin]

Ok thanks

[gray stone]

You just need the formulas, really.

[halcyon ruin]

Would they work for these questions too?

[gray stone]

Ah, although, hold on...

Actually, I don't think the usual formulas will work here, since we only have a limited number of each type of object.

[halcyon ruin]

Oh

So there’s no quicker way to do these

[gray stone]

Well, there might be a way to generalize this, but I can't say I remember how to do that.

[halcyon ruin]

Would u be able to help me with question 16 ii please

[gray stone]

This could help.
https://math.stackexchange.com/a/554237

[gray stone]

Oh, easy enough: find the number of arrangements of 9 cars, then multiply that by the number of options to place three consecutive empty spaces.

[halcyon ruin]

Yeah that gave me the right answer thanks

Wait

Nvm

Makes sense thanks

[halcyon ruin]

Also with binomial distribution how do you know when to use it rather than condition probability?

[gray stone]

Not really sure how exactly those two are related.

Binomial distribution is used when you have n independent trials with the same probability of success p. Then the amount of successes is a random variable X ~ Bin(n, p).

[halcyon ruin]

Alright thanks

Also have another question about a different topic should i create a new thread?

[ivory comet]

what book is this from, just curious i feel like ive seen it before

[halcyon ruin]

OCR additional maths

[ivory comet]

i did that

i hated perms and combs bro

[halcyon ruin]

How was it

[ivory comet]

so glad it didnt come up thay much in my exam

[halcyon ruin]

Do u understand it now?

[ivory comet]

easy

[halcyon ruin]

I’m going through that struggle rn lol

[ivory comet]

nah, not too much but im doing it in furtherpure 2

so il have to learn it eventually

[halcyon ruin]

What other topics did u find hard for it

[ivory comet]

just perms and combs

[halcyon ruin]

Have my mock paper tomorrow

[ivory comet]

the rest is fine il be honest

u should be good

[halcyon ruin]

Alright thanks

[ivory comet]

are u fine with all ur formulas

trapezium rule n stuff

[halcyon ruin]

Need to go over that

[ivory comet]

fair

log rules

integration and differentiation are fine probably for you too?

[halcyon ruin]

Yeah I’m decent at that id say

[halcyon ruin]

Yep

[ivory comet]

the trig is messy

did u have a look at past papers

[halcyon ruin]

Can u do most of the stuff that’s on the paper w Casio cg50

[halcyon ruin]

Trig equations u can do it on that

[ivory comet]

i didnt have one at the time

[ivory comet]

ye

its useful but i use CAST diagrams

[halcyon ruin]

Was just gonna have a look now

[halcyon ruin]

What’s this?

[ivory comet]

so

il send u a video, its really good for finding all values

i use it instead of the calc ngl

[halcyon ruin]

Dyk anything else that I can use cg50 for apart from solveN function and graphs

[halcyon ruin]

For trig?

[ivory comet]

https://youtu.be/Amd1pbHj_Ic?si=yvhqAZaHkfyIgglt

ye

for sine cos tan

[ivory comet]

hmm lemme think

i use the small one more

fx991 or something

i forgot

white back

like cg50 i only use for graphs il be real

[halcyon ruin]

What’s the point of this if u can just put it into the calculator tho?

Of the cast diagram

[ivory comet]

does calc give all values

not just principle

[halcyon ruin]

Oh yeah

It doesn’t give all values

[ivory comet]

i dont know if it does, i dont use calc

[ivory comet]

cast finds all values

in a given range

[halcyon ruin]

Oh ok seems abit too long to learn rn tho icl

[ivory comet]

yea, u prolly wanna focus on ur weaknesses

or stuff u dont remmber as much

u remember ur trig identities?

[halcyon ruin]

Yeah

Sin squared + cos squared

= 1

[ivory comet]

ye

and one more

[halcyon ruin]

Sin / cos = ram

Tan

[ivory comet]

yessir

what else is there

linear programming

[halcyon ruin]

How much harder is a level maths than additional maths

Is it like way harder or

[halcyon ruin]

Is this just using logs to get it into the form y=mx + c

[ivory comet]

idk man alevel is boring for me

finished most of it for maths and fm

just got a module left

[halcyon ruin]

Fairs

[ivory comet]

i dont completely remember, but isnt that something else

[gray stone]

Linearization.

[ivory comet]

im not really too sure but i do remember the thing with the y=mx + c logs thing

[gray stone]

Used extensively in chemistry and physics.

[ivory comet]

ye but i forgot the topic name

in the book

[halcyon ruin]

So what’s linear programming ?

Plotting the graphs?

[ivory comet]

make sure its in ur book first

[gray stone]

I've heard of it, but I don't know.

[halcyon ruin]

Yeah it is

Wait

It’s called reduction to linear form

Is that the same or?

[ivory comet]

thats the y=mx + c stuff then

[halcyon ruin]

Ohh alr

[ivory comet]

linear programming is somethjng different

[gray stone]

Yeah, that's linearization.

[ivory comet]

if its not in ur book maybe ur doing a diff level or something, im not sure

[halcyon ruin]

Is it like linear regression

Or is that something completely different

[ivory comet]

send a pic of the contents rq? if u dont mind

[halcyon ruin]

Alr

[gray stone]

Nah.

[gray stone]

A lot of stuff is linear, but it's not the same.

[ivory comet]

page 78

is your mock paper on everything?

or just specific topics

[halcyon ruin]

Ohh linear inequalities

[halcyon ruin]

Yeah everything

[ivory comet]

thats another way to put it ye

u got that one down?

[halcyon ruin]

Can do that on graphing calculator I’m pretty sure

[ivory comet]

oh shit fr?

damn thays useful

[halcyon ruin]

I think

If u just put in the equations

[ivory comet]

cuz u needa find the minimum and maximum val for x and y

or stuff like that

[halcyon ruin]

It shows u the region

[halcyon ruin]

Then solve for this using g solve function

[ivory comet]

ur using ur calc to its full potential lmao

i havent heard of any of this but its good to know

[halcyon ruin]

Working out marks tho

[ivory comet]

yk linear programming is a further maths decision alevel topic

idk why its in ocr admaths

its not hard but its boring

[halcyon ruin]

Not even in normal a level maths?

[ivory comet]

polynomial division is a lot of marks i think

iirc

[halcyon ruin]

Do u need to know long division

Polynomial long division

[ivory comet]

polynomial long division yea

u need everything in the book if ur test is on that

pld is easy tho

[halcyon ruin]

I’m good at grid method

[ivory comet]

i mean if check the book

itl tell u what method to use

[halcyon ruin]

Yeah it is

[ivory comet]

normal alevel maths is so dead

boring af

[halcyon ruin]

But would questions ask to do it with a certain method?

[ivory comet]

honestly i only use the long division

[halcyon ruin]

Did u have to pick alevel maths to do further maths

[ivory comet]

its fast and efficient

[halcyon ruin]

Oh

[ivory comet]

ye

[halcyon ruin]

Might go over that then asw

[ivory comet]

further maths is an extension to normal maths

so u cant do fm without normal maths

[gray stone]

Wait, you learn linear programming in school?

[ivory comet]

ye

[gray stone]

Isn't that concerned with minimization of multivariable functions?

[ivory comet]

its a qualification i did and hes doing

called additional maths

[ivory comet]

just x,y so its not that bad

[halcyon ruin]

z gets involved later on too?

[gray stone]

Well, still, two variables. Something more like university-level stuff.

[ivory comet]

no idea

[ivory comet]

its not too bad honestly

like

we learn the basics

[gray stone]

Hm... Well, probably because the functions are linear.

[ivory comet]

and people who do normal maths never hear of it again lmao

sadly im doing decision maths so i do hear of it again next year

[gray stone]

I mean that in general optimization of multivariable functions is a topic in calculus.

[ivory comet]

ye

i despise decision maths

[gray stone]

And this is constrained optimization, too...

[ivory comet]

its like graph subgraph group eularian circuit

[gray stone]

So, in general, Lagrange multipliers and bordered hessian, which is quite a pain.

[ivory comet]

probably not something with all those fancy words

its quite simple

at this lvl

cuz its for like

14-15 year olds

or 15-16

prolly 14-15

[gray stone]

I see.

Well, it's probably a lot easier, since from what I've seen not only the function is linear, but so are the constraints.

[ivory comet]

ur fine with kinematics?

[halcyon ruin]

@gray stone can u do linear programming on graphing calculator?

[ivory comet]

variable acceleration

[halcyon ruin]

Suvat stuff

[ivory comet]

variable

specifically

just think SVA

[gray stone]

As I said, I haven't learned it, so I don't know. I believe it's usually done on a computer.

[halcyon ruin]

Might need to revisit that again

[ivory comet]

rightways is differentiate

[halcyon ruin]

Oh alright

[ivory comet]

integrate is leftways

[halcyon ruin]

Yeah and u do it twice to go from

I think acceleration to distance?

[ivory comet]

s= displacement
v= velocity
a=acceleration

down differentiate
up integrate

when acceleration is variable ie the questions are a function in terms of t like “the displacement of a particle is t^2 + t^3 find acceleration”

[ivory comet]

second derivative of displacement is acceleration yes

[halcyon ruin]

And you’d need the differentiate twice here?

[ivory comet]

yes

[halcyon ruin]

And u get

6t?

[ivory comet]

u should also know about how to differentiate for the turning point

[gray stone]

Turning point?

[ivory comet]

how to find if a TP is a minimum or a maximum

[halcyon ruin]

Make it = 0

[gray stone]

I've heard this term before. What does it mean, exactly?

[halcyon ruin]

And sub values back in

[ivory comet]

0,0 for x^2 example

its where the graph turns basically

not a point of inflection

dy/dx = 0

[gray stone]

Oh, you mean extremum point?

[ivory comet]

ye

we call it tp here

[gray stone]

Hm...

[ivory comet]

or minimum/maximum

[gray stone]

I don't really like that name.

[halcyon ruin]

Is this right?

[ivory comet]

lemme check

[gray stone]

Well, critical points are still called critical, right?

[ivory comet]

t^2 + t^3
2t + 3t^2
2 + 6t

2+6t = a

[ivory comet]

wdym by critical

[halcyon ruin]

Oh yh forgot to keep the + 2

[ivory comet]

yep

how do u know if 2+6t is minimum or maximum at t=3?

and then do the same test for -10

[gray stone]

Critical points are where the derivative equals zero or doesn't exist. Extremum points are critical points that are either maximum or minimum points.

(replace derivative by gradient for multivariable functions)

[halcyon ruin]

Check the y value right

[ivory comet]

i dont remember the name exactly, cuz we find if its an extreme point or if it inflects

[ivory comet]

just substitute

[halcyon ruin]

Sub back into equation

[ivory comet]

2+6t at t=3

[halcyon ruin]

See which value is lower

[ivory comet]

is

2+6(3)

20>0 therefore minimum

[gray stone]

Well, the usual process is that you find critical points, then test which ones are extremum points (and what type) and which aren't.

[halcyon ruin]

Min or max?

[ivory comet]

in general
if f’’(t)>0 minimum point
f’’(t)<0 max

[gray stone]

If f''(t) exists and not equal to zero, at least.

[ivory comet]

f’’(t) represents 2nd derivative at the point t=t

[ivory comet]

ye

in this case check the left and right side

of t

in small intervals

[halcyon ruin]

Ibr id just check on graphing calculator

[ivory comet]

its weird ik but its kinda the opposite

[gray stone]

So, for example, f(x) = x^3 (1 - x) has two critical points: x = 0 and x = 3/4. But only x = 3/4 is an extremum point (more specifically, a maximum point), while x = 0 isn't an extremum point.

[ivory comet]

some graphs get nasty

[gray stone]

Or if the function is sufficiently smooth, you can also check higher-order derivatives. Useful for polynomials.

[halcyon ruin]

Even in add maths?

[ivory comet]

ur gonna get stuff to the power of 1/3 and 3/5 etc

not that difficult if u know ur content

but ur wasting time if u plug it into a graphing calculator

without knowing WHY ur doing it

[halcyon ruin]

Hmm yeah

[ivory comet]

graphing calculators i feel like are more of a tool to check ur work

[gray stone]

Yeah, true.

[ivory comet]

ion even use mine apart from when im working with polar curves

[halcyon ruin]

Isn’t it easier sometimes to work backwards tho

[ivory comet]

cuz its useful to understand why an equation outputs said graph

[gray stone]

Well, you won't be able to do that on a test 😅

[ivory comet]

if it works for you, go for it

idk if ur allowed cg50 graphing in admaths

[gray stone]

You need to do it by yourself, too.

[ivory comet]

i didnt see anyone with one in my exam

only a normal calculator

[gray stone]

Wait, calculator? On a math exam?

[ivory comet]

ye

[gray stone]

Huh, that's weird...

[ivory comet]

in the qualification its calculator

in the actual exam gcses

[gray stone]

Why is that allowed? And how would it help, anyway?

[ivory comet]

its 1 non-calc 2 calc

[gray stone]

Oh, wait.

It's a school exam.

[ivory comet]

ye

[gray stone]

Yeah, then it might help. Though, I still find it strange that it's allowed.

I've never seen a calculator allowed on a math exam, and it would be useless in a uni math exam, anyway.

[ivory comet]

i mean ion think it helps too much but at the same time timed conditions are annoying

[gray stone]

Anyway, it's quite late for me now, so let's continue this conversation in the afternoon or evening.

[ivory comet]

i always find myself with more time than i need though, so maybe no calc is the way to go lmao

okie dokie

[halcyon ruin]

@ivory comet binomial distribution

[ivory comet]

that ones quite easy

[halcyon ruin]

When it says atleast one condition

[ivory comet]

havent done it in a while send me a question type

and i can explain the question

[halcyon ruin]

So 4 ordinary dice are thrown

1-6

[ivory comet]

x~B(n,p)

[halcyon ruin]

Find probibility atleast one three is rolled

[ivory comet]

do yk what this means

[halcyon ruin]

Yeah

n is number of trial

[ivory comet]

fill it out

with your question

[halcyon ruin]

P is probiliblty of success

[ivory comet]

last thing before i go im tired

[halcyon ruin]

X~B(4 , 1/6)

[ivory comet]

ye

[halcyon ruin]

Alright

[ivory comet]

so whats the question again lol

[halcyon ruin]

4 dice thrown numbered 1-6

Probability of atleast one three

[halcyon ruin]

Six sided dice

[halcyon ruin]

U find p of 0 threes right?

Then do 1 - that

[ivory comet]

surely u dont need binomial for this

[halcyon ruin]

Huh

What’s the point of it then

[ivory comet]

oh im dumb

i read the qn wrong

mb

its

[ivory comet]

this

whatd u get in the calculator for this?

[halcyon ruin]

But what do u do after that

Idk how to put it into graphing calcualtor

Lol

Should I take both to exam?

[halcyon ruin]

991 and this

[ivory comet]

dont u know the formula

for bin dist

[halcyon ruin]

It’s like something c domething times p of success ^

[ivory comet]

ncr * p^r * (1-p)^n-r

[halcyon ruin]

Yeah

[ivory comet]

do u know ur binomial expansion too

[halcyon ruin]

Yeah

[ivory comet]

that ones quite good all sorts of questions come up for that

(x+2/x)^6

find the constant term if one exists

[halcyon ruin]

Wait what does r mean again

[halcyon ruin]

I just did that exact question

[ivory comet]

p is probability
n is total trials
r is trials ur looking at

[halcyon ruin]

Thanks

[ivory comet]

whays the constant term

then

[halcyon ruin]

160

[ivory comet]

is it the exact

same one

wtf

[halcyon ruin]

Yh lol

[ivory comet]

cuz i jst madd that one up

LOL

[halcyon ruin]

Crazy

[ivory comet]

whay about

(x+3/x)^5

find if it exists, if not

if doesnt exist prove why not

prove it doesnt exist

[halcyon ruin]

X^-3 can’t be times by something in that equation to = x^0

There’s no x^3 term

[ivory comet]

sum from 5 to 0 of x^r * (3/x)^5-r * 5cr
(x^r )( x^r-5) = x*0

2r - 5 = 0

r = 5/2

r has to be a whole number

[halcyon ruin]

Huhh

[ivory comet]

hmm

u havent done sums?

[halcyon ruin]

A bit

[ivory comet]

this is the way i was taught these sorta questions

[halcyon ruin]

Not much

[ivory comet]

write this down

[halcyon ruin]

Ok

[ivory comet]

the sum = (a+b)^n

in this case, a=x and b=3/x and n=5

correct?

[halcyon ruin]

Yeah

[ivory comet]

.

alr write that down using the sum formula

the constant term is when x^0 correct?

[halcyon ruin]

Yh

[ivory comet]

because itl have no x

next to it

right, so we want to find the term in our sum when x^0

so ignoring everything else, extract the
x^r * (1/x)^n-r

cuz the nCr and the 3 dont matter

they are constants right

no x term

[halcyon ruin]

Yh

[ivory comet]

(1/x)^n-r

how do you flip this?

[halcyon ruin]

Times -1?

Wait

[ivory comet]

ye

[halcyon ruin]

Yh

[ivory comet]

so whats it now

[ivory comet]

this turns into x^r * (x^r-n)

[halcyon ruin]

X^-n-r

[ivory comet]

are u happy up to here

[ivory comet]

-(n-r)

[halcyon ruin]

Would something like this come up tho?

[ivory comet]

i did some questions on it ye

its up to u

dependso n ur teachers

ion mind teaching it if u think ur teachers wont put it on the mock

[halcyon ruin]

Leave it I think it should be fine

I think I’d be better off learning other stuff rn

[ivory comet]

defo perms and combs lmao

[halcyon ruin]

Thanks tho

[ivory comet]

ts is so annoying

[halcyon ruin]

Tm

[ivory comet]

u got this, gl

im gonna go now

[halcyon ruin]

Thanks

[ivory comet]

bye bye

[halcyon ruin]

Bye

[stoic ether]

@halcyon ruin

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