Can anyone please explain The Binomial Theorem and Pascal’s Triangle to me? I will medal?!

- anonymous

Can anyone please explain The Binomial Theorem and Pascal’s Triangle to me? I will medal?!

- jamiebookeater

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- anonymous

Practice problems would be great too because I have no idea how to do it

- ganeshie8

First of all, recall that a "binomial" is a polynomial with "two" terms. so the theorem must have something to do with "two terms".

- anonymous

Ok:)

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## More answers

- ganeshie8

Indeed it does, the binomial theorem gives a nice expression for repeated multiplication of a binomial with itself:
\[(x+y)(x+y)(x+y)\cdots\]

- anonymous

I see, I see

- ganeshie8

Without beating around the bush, here is the theorem in all its glory :
\[\large (x+y)^n = \sum\limits_{k=0}^n \color{red}{\binom{n}{k}}x^ky^{n-k}\]

- anonymous

I'm just gonna go ahead and do pascals triangle.
In very simple words, its a number triangle.
To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern.
Each number is the two numbers above it added together (except for the edges, which are all "1").

- anonymous

:O Stop! The formula was NOT what I was expecting...

- anonymous

But how does it relate to the binomial theorem

- ganeshie8

\(\large \color{red}{\binom{n}{k}}\) is called "binomial coefficient", pascal's triangle helps you in finding this coefficeint.

- anonymous

Let’s explore the first couple of powers of the binomial (a + b).
(a + b)0 = 1 *Remember, anything to the zero power equals 1.
(a + b)1 = a + b
(a + b)2 = (a + b)(a + b) = a^2 + 2ab + b^2
(a + b)3 = (a + b)(a2 + 2ab + b2) = a^3 + 3a^2b + 3ab^2 + b^3

- anonymous

uh-huh I'm processing this slowly, but I feel like I'm getting it

- anonymous

Now take a look at the pascals triangle.
(just search it up online)

- anonymous

compare the coefficients, and powers with the pascals triangle

- anonymous

1
a + b
a^2 + 2ab + b^2
a^3 + 3a^2b + 3ab^2 + b^3

- anonymous

and
1
11
121
1331

- anonymous

Yes, I understand that.

- anonymous

Can any of you guys give me like a practice question to help maybe?

- anonymous

The row represents the coefficients

- anonymous

Hold on ffggff i just need to say one more thing

- anonymous

kk:)

- anonymous

Take a look at the exponents for (a+b)^3 The powers of a decrease from 3 to 0, and the powers of b increase from 0 to 3.

- anonymous

so that is the relationship between the two
Here is a practice problem
Expand (x + 2)^5 using the Binomial Theorem and Pascal’s triangle.

- anonymous

Side note, is that the fifth row of pascals triangle would represent (a+b)^4, and the third would be (a+b) ^2. The reason is that the first row, represents (a+b)^0

- anonymous

Ok, I'm lost lol. So I drew the little triangle thingy and got 1, 5, 10, 10, 5, 1. But now what?

- anonymous

Like I can do the triangle but I don't know what to do afterwards?

- anonymous

The sixth row of Pascal’s triangle would be 1, 5, 10, 10, 5, 1 and the exponents would start and end with 5. And that is what you got

- anonymous

Yup

- anonymous

So then
\[1x^5 2^0 + 5x^4 2^1 + 10x^3 2^2 + 10x^2 2^3 + 5x^12^4 + 1x^02^5 =\]
\[x^5 + 5 • 2x^4 + 10 • 4x^3 + 10 • 8x^2 + 5 • 16x + 32 =\]
\[x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32\]

- anonymous

:O Whoa! Okay! How did you add the 6th row of the triangle to the equation?

- anonymous

I mean, how did you get those exponents?

- anonymous

Remember this? Take a look at the exponents for (a+b)^3 The powers of a decrease from 3 to 0, and the powers of b increase from 0 to 3.
SO basically, as it goes, the powers of x decrease from the n to 0 (where n is the exponent, or (x+y)^n) And the powers of the other one (in this case it is 2) increases from 0 to n.

- anonymous

Ok :) Thank-you, I get it now!!!!

- anonymous

NP!!! :)

- anonymous

So basically they are coming from (a+b)^5 in a way?

- anonymous

@zzr0ck3r I feel so bad! Lol you have been typing forever :)

- anonymous

well, yes,
So to finalize it let the equation be (a+b)^n
So a's exponents would go from n to 0 and b's would increase from 0 to n.

- anonymous

how do you what to do to break it up into 5? Like how the first one is 1x^5 2^0 and the second one is 5x^42^1?

- zzr0ck3r

A neat trick you can use instead of the triangle.
Example:
\((a+b)^4\)
We know that if will look something like this
\(\text{_}a^4+\text{_}a^3b+\text{_}a^2b^2+\text{_}ab^3+\text{_}b^4\)
Now the question is, what do we put in the _ places
Well we know the first coefficient will be \(1\).
For the second _ we multiply the power of \(a\) in the first term, by the coefficient in the first term and then divide that all by \(1\) (we use one because the previous two steps were in the first term). So \((4*1)/1=4\) and that is our second coefficient.
so we have \(a^4+4a^3b+\text{_}a^2b^2+\text{_}ab^3+\text{_}b^4\)
For the third coefficient, we multiply the power of \(a\) in the second term by the coefficient in the second term and divide by \(2\) (we are doing stuff in the second term now) and we get \(3*4/2 = 6\) and that is our third coefficient.
Similarly our fourth coefficient is \(2*6/3=4\) and our last coefficient is \(1*4/4=1\)
So we have \(a^4+4a^3b+6a^2b^2+4ab^3+b^4\). Also I am sure someone pointed this out above, but once we find the first half of the coefficients, we are done (notice the pattern).
Ok this all seems very hard and long, but that was just because I was doing lots of steps.
If wanted to do \((a+b)^7\)
First we quickly write out
\(a^7+a^6b+a^5b^2+a^4b^3+a^3b^4+a^2b^5+ab^6+b^7\)
and note the leading coefficient is \(1\).
then we do
\(7*1/1=7\\
6*7/2=21\\
5*21/3=5*7=35\)
and we are done because we have gone half way
so we have \((a+b)^7=a^7+7a^6b+21a^5b^2+35a^4b^3+35a^3b^4+21a^2b^5+7ab^6+b^7\)

- anonymous

do you see what I mean kinda?

- zzr0ck3r

I did not realize it was going to be so long once I started...

- anonymous

@zzr0ck3r Thank-you :), I might just use your way from now on. It's simple yet very effective!

- anonymous

well yes @zzr0ck3r but ffggfgf wants to know the relationship between binomial theorem and pascals triangle.

- zzr0ck3r

this actually shows it:)

- anonymous

It's a good method.

- zzr0ck3r

I can generally do these things and write them down faster than someone can put it in a calculator and write it down.

- anonymous

i use a different method than these, it is quite fast also.

- zzr0ck3r

But anyway, it was meant to be a little extra add-on to what you all were saying...

- anonymous

i would write it, but i don't want to spend another hour lol

- anonymous

Thanks to everyone who helped!! I really understand how to do it now!! :)

- ganeshie8

I think that works because of combinations, which term we pick in each binomial :
|dw:1436843111371:dw|

- misty1212

one day see if you can find a nice proof of why pascal's triangle works for these questions, and also why the construction of the triangle give the entries \(\binom{n}{k}\)
someday later on i mean

- anonymous

Thanks misty1212 :) Lol once I get to Ap cal I will!

- anonymous

Then what do i do ganeshie?

- ganeshie8

You will know what to do once you're faced with an example problem.
Let me see if I can cookup an interesting problem :)

- anonymous

Kinda scared now hahah :) But I will try

- ganeshie8

Find the coefficient of \(x^6\) when you expand out below product :
\[(x+1)(x+1)(x+1)(x+1)(x+1)(x+1)(x+1)\]
(there are 7 terms in the product)

- anonymous

Ok, hold on please

- anonymous

Would it be 1?

- ganeshie8

nope, but how did you guess 1 ?

- anonymous

1 9 36 84 126 126 84 36 9 1

- anonymous

I did the (a+b)^6 using the triangle

- anonymous

the first one would be one, as the first one would be x^7.

- anonymous

remember what i said the exponents on the x part decrease from n. And btw, @ganeshie8 typed the expanded version of (x+1)^7

- anonymous

ohhh whoops But yes, go on.

- ganeshie8

Yes, maybe forget about the theorem/pascal's triangle and try working it pretending we never heard of these

- anonymous

tru tru, lets forget lol

- dan815

@misty1212 you wanted a proof for why the construction of pascal triangles has combinatoric solutions?

- ganeshie8

\[(x+1)(x+1)(x+1)(x+1)(x+1)(x+1)(x+1)\]
Notice that there are \(7\) factors here and since we want the coefficient of \(x^6\), we must pick "x" from "6" factors and pick "1" from one factor.
We can pick "1" from any of the 7 factors in exactly 7 ways. so the coefficeint of \(x^6\) in above product is simply \(7\)

- anonymous

Ok!

- ganeshie8

As you can see, the "coefficient" is really equivalent to "number of ways of choosing a factor"

- anonymous

so would it be 1 7 21 35 35 21 7 1

- anonymous

Yup!

- ganeshie8

If you use pascal's triangle, yes.

- anonymous

Gotcha

- ganeshie8

Let me ask you a side question :
Suppose you went for shopping and liked 7 different shirts,
but you can only buy 6 shirts. how many total ways are there to choose 6 shirts from 7 shirts ?

- dan815

|dw:1436844308425:dw|

- anonymous

6

- ganeshie8

think again

- anonymous

6x7 basically

- ganeshie8

Let me change the question slightly :
There are 7 shirts and you can buy only 1 shirt. How many ways are there to choose 1 shirt from 7 shirts ?

- anonymous

7? I don't really understand the question. Can u just explain it to me please?

- ganeshie8

Correct! I will explain after the drill in the end. lets try another related question :
There are 7 shirts and you can buy only 1 shirt. How many ways are there to "reject" 6 shirts from 7 shirts ?

- anonymous

7?

- ganeshie8

Yes you do see that "choosing 1 shirt from 7 shirts" is same as "rejecting 6 shirts from 7 shirts"

- anonymous

yup. I do see.

- anonymous

At first I didn't, but now I do.

- ganeshie8

"choosing 1 girlfriend from available 7 friends" is same as "rejecting 6 girls from the available 7 friends"

- ganeshie8

sry for bad analogy but you get the point

- anonymous

Lol yeah, I get it :D

- ganeshie8

so what has that anything to do with finding the coefficient of \(x^6\) in the below product :
\[(x+1)(x+1)(x+1)(x+1)(x+1)(x+1)(x+1)\]
?

- anonymous

oh, okay so I'm assuming it would be 7

- anonymous

Yes! Omg! Ok, I see. Visuals really do help!!

- ganeshie8

Maybe writing the product pictorially helps in seeing whats going on :
|dw:1436845412826:dw|

- ganeshie8

I feel that is enough for one session, just open a new thread if you still have any question :) good luck!

- anonymous

Thanks for all your help<3

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