Sequence, Binomial Expansion /Theorem and Pascal's Triangle

[sly eagle]

Please someone help me answer this.

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[fading adder]

okie, did you do the first one? (x+y+z)^3 yet ?

[sly eagle]

No, Im still stuck there for hours

[fading adder]

okie lets do that then

question says take x as a and y+z as b

(a+b)^3 then

which we can use binomial expansion upon ?

@sly eagle do you know how to use binomial expansion 👀

[sly eagle]

no i have no idea, my teacher did not discuss either of these

[fading adder]

oo okie then lets start there

$(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

@sly eagle lets try this on our question, what is n in our question ?

[sly eagle]

uh 3??

[fading adder]

okie lets put it in then

$\sum_{k=0}^{3}\binom{3}{k}a^{3-k}b^k$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

what is 1st term gonna be ?

[sly eagle]

idk

[fading adder]

for 1st term, we put k = 0

and find whatever it equals inside the sum

$\binom{3}{0}a^3b^0$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

thats the first term

2nd term will have k =1

then 2 then 3

[fading adder]

do you know how to expand that binom ?

[sly eagle]

nope

[fading adder]

ah ok

$\binom{n}{k}=\frac{n!}{k!(n-k)!}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

do you know what factorials are ?

[fading adder]

what n! means

[sly eagle]

no

[sly eagle]

nope 😓

[fading adder]

[sly eagle]

😭

[fading adder]

okie okie lets start with factorials then

so lets say you have 3!

it means you multiply all integers till 3

(1)(2)(3)

whats 4! ?

[sly eagle]

(1)(2)(3)(4)??

[fading adder]

yup

[fading adder]

now look at the binom formula

we are gonna be applying that

$\binom{3}{0}a^3b^0$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

this is our first term

[fading adder]

also thing to note 0! is 1

[sly eagle]

oh okays okays

[fading adder]

you will learn later why and how

for now just take my word on it

[fading adder]

what would you get when you expand this binom 👀

[sly eagle]

😨 idk

[fading adder]

okie imma show you an example then

and then you will do this one and next ones

$\binom{n}{k}=\frac{n!}{k!(n-k)!}$ this is our formula

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

and we will find $\binom{4}{3}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

so lets put it in our formula

$\frac{4!}{3!(4-3)!}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

1! is just 1

$\frac{4!}{3!1!}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

so $\frac{4!}{3!}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

$\frac{(4)(3)(2)(1)}{(3)(2)(1)}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

and what does this equal 👀

@sly eagle

[sly eagle]

uh 24/6?????

[fading adder]

(3)(2)(1) can be cancelled in denominator and numerator right ?

[fading adder]

you can multiply and then divide like t his

[sly eagle]

I guess?

[fading adder]

so what are you left with ?

[sly eagle]

4

[sly eagle]

where did 1 go here though

[fading adder]

yup

[fading adder]

1 was in multiplication right ?

so it doesn't matter really

[sly eagle]

oh okays okays

so is the answer 4?

[fading adder]

yup

now lets get back to our question

$\binom{3}{0}a^3b^0$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

b^0 is 1

expand $\binom{3}{0}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

and you will have your first term

[fading adder]

what would you get @sly eagle 👀

[sly eagle]

wait

[fading adder]

okie

[fading adder]

I gtg for lunch

I will be back soon

[sly eagle]

I got (3)(2)/(0) 😭

[fading adder]

try finding all 4 terms till then

[fading adder]

0! = 1

[sly eagle]

i got (3)(2)/(1) then??

[fading adder]

$\frac{3!}{0!(3-0)!}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

nope

$\frac{3!}{3!}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

so 1

[fading adder]

3-0 = 3

now imma brb

[sly eagle]

okays okays

[fading adder]

I am back @sly eagle

[sly eagle]

how did it become 3 in the end

[fading adder]

oo final answer isn't 3

[fading adder]

this became 1 in the end

[sly eagle]

im confuse

[fading adder]

oo okie

$\frac{3!}{0!(3-0)!}$

[warm mural]

BabyYoda#gtfoanemia

[sly eagle]

oh

[fading adder]

$\frac{3!}{1*(3)!}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

right ?

$\frac{3!}{3!}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

= 1

[sly eagle]

why it did not become (3) (2) (1)??

[fading adder]

well we can do that

$\frac{(3)(2)(1)}{(3)(2)(1)}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

but when we cancel it out its gonna be 1 anyhow

[sly eagle]

okay okay

[fading adder]

okie so 1st term is $a^3$ then ?

[warm mural]

BabyYoda#gtfoanemia

[sly eagle]

yes?

[fading adder]

now lets go for 2nd term

$\sum_{k=0}^{3}\binom{3}{k}a^{3-k}b^k$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

last time k=0

this time in 2nd term k = 1

and 1st and 2nd terms will be added together

same for all terms that come ahead

[sly eagle]

okay

[fading adder]

what would the 2nd term be 👀

try working it out

first expand the binom

then we can figure out a and b later

[sly eagle]

is the 2nd term kbk?

[fading adder]

kbk ☠️

k=1 ?

so do you mean b ?

[sly eagle]

yeah

[fading adder]

nah b is not the 2nd term

[sly eagle]

[fading adder]

first we gotta solve $\binom{3}{1}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

try doing that

[sly eagle]

oh

[fading adder]

did you solve it 👀

[sly eagle]

is it 3???

[fading adder]

yup

good job

[sly eagle]

okays okays

[fading adder]

now that we have that, lets find full term

$\binom{3}{1}a^{3-1}b^1$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

what does this equal ?

[sly eagle]

idk 😭

[fading adder]

uhm

$3a^2b$

[warm mural]

BabyYoda#gtfoanemia

[sly eagle]

im confuse

how did it become like that

[fading adder]

you said $\binom{3}{1}$ = 3 ?

[warm mural]

BabyYoda#gtfoanemia

[sly eagle]

yes

[fading adder]

3-1 = 2, thats why $a^{3-1} = a^2$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

and $b^1$ is just b

[warm mural]

BabyYoda#gtfoanemia

[sly eagle]

oh

[fading adder]

so this became $3a^2b$

[warm mural]

BabyYoda#gtfoanemia

[sly eagle]

oh okay

[fading adder]

did you understand what happened here ?

[sly eagle]

I guess so

[fading adder]

now the 3rd and 4th term you have to solve all by yourself

so if any doubts, now is the time to ask

[sly eagle]

is the 3rd term (3/2) and 4 is (3/5)??

[fading adder]

3/2 and 3/5 ?

wdym

[sly eagle]

wait idk how to type it

[fading adder]

use $\binom{a}{b}$

put it there

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

where I put a and b

[fading adder]

why $\binom{3}{4}$ ?

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

and not $\binom{3}{3}$

[sly eagle]

i mean 3

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

yup

[fading adder]

thonk

[fading adder]

and what about a and b ?

[sly eagle]

huh what a and b

[fading adder]

what will their powers be in 3rd and 4th term 👀

[fading adder]

in this we have a and b right ?

[sly eagle]

yes

[fading adder]

this is where you get a and b form

[sly eagle]

oh

wait

[fading adder]

👀

[sly eagle]

3a2b?? for the 3rd term??? 😓 😭

[fading adder]

wasn't that the 2nd term ?

[fading adder]

here

[fading adder]

you are close, but far

look again you can do it

[sly eagle]

i got this wait

[fading adder]

why did you take k =1 ?

[sly eagle]

😭

[fading adder]

you took k = 1 at 1 place

and k = 2 at once place ??

[sly eagle]

now I'm confused

[fading adder]

where did you get $\binom{3}{2}$ from ?

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

$\sum_{k=0}^{3}\binom{3}{k}a^{3-k}b^k$ it comes from this right ?

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

we put k = 2

[sly eagle]

Cause like you did the (31) thingy so I thought it just like that

😓😓😓

[fading adder]

where does (31) thingy come from tho ?

[fading adder]

all of it comes from this

[sly eagle]

the this one wait

[fading adder]

you put values in, from 0 to 3

we put 0 and 1 , now we put in 2

[fading adder]

yeshh thats the spirit

gl

[sly eagle]

the that ine

[fading adder]

wha

[sly eagle]

idk how to explain how i understood it

[fading adder]

(3 1) ?

[sly eagle]

yeah that one

when u put 3 from above then 1 from lower

[fading adder]

yes, 1 from lower was because we put k = 1

its actually $\binom{3}{k}$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

and we put whatever k is inside there

$\sum_{k=0}^{3}\binom{3}{k}a^{3-k}b^k$

[warm mural]

BabyYoda#gtfoanemia

[fading adder]

k starts from 0 and goes till 3

[sly eagle]

Hi, I am just gonna close this. I just found that task was not our home work Still thank you so so much, as much I still wanna ask about them for advance study I still have some resarch rrl need to do 😓 😭 I am really sorry to for wasting your time. I really am.

[sleek sable]

@sly eagle has given 1 rep to @fading adder

[sly eagle]

Thanks you too for your patience with me

+close