Question

What is the third term of (x+y)^9

Answer 1

\[\left( x+y \right)^9\]

Answer 2

\[\large\rm (x+y)^9\quad=\sum_{k=0}^9 \left(\begin{matrix}9 \\ \rm k\end{matrix}\right)x^{9-k}y^k\]Binomial Theorem, ya?
Hmm no coefficients on the x or y, that makes this one a bit easier :))

Answer 3

Oh you're still typing, hehe.
I'll wait ^^

Answer 4

Possible answers are:

A: \[84x^6y\]
B: \[84^7y^2\]
C: \[36x^6y^3 \]
D: \[36x^7y^2\]

Answer 5

I apologize for taking a while, math's not my strong suit. Working on improving it though.

Answer 6

We start counting from k=0, right?
So we count 0, 1, 2... k=2 is going to be our third term.

\[\large\rm third~term\quad=\left(\begin{matrix}9 \\ 2\end{matrix}\right)x^{9-2}y^2\]

Answer 7

So I multiply that?

Answer 8

Yes, simply the exponents.
To deal with the 9 choose 2, you can rewrite it in terms of factorials,
or if you don't mind the hassle, you can go down to the 10th row of pascals triangle

Answer 9

This is one option :)\[\large\rm \left(\begin{matrix}n \\ \rm k\end{matrix}\right)=\frac{n!}{k!(n-k)!}\]

Answer 10

So I put x and Y in the k and n sections and then multiply it by the 9 which is in the n! and k! sections?

Answer 11

ummmm 0_o

Answer 12

Oh, I messed up slightly, A is supposed to be \[84x^6y^3 \]

Answer 13

And now I'm confused at where to put the numbers and variables in.

Answer 14

Hmm ok, if Binomial Theorem is confusing,
maybe you can do it another way.

Just realize that the first thing in the brackets (x in this problem)
starts at the largest power of 9, and counts down in each term.

While the other thing starts at nothing, and counts up to 9.

\[\rm (x+y)^9=\text{__}x^9y^0+\text{__}x^8y^1+\text{__}x^7y^2+\text{__}x^6y^3+...\]With a bunch more terms to follow, but we don't need those.

Answer 15

I left a blank in front of each term,
that is the coefficient coming from Pascal's Triangle.

Answer 16

So it would be either A or C then.

Answer 17

`third` means... three.
yes?
So are you sure you counted the `third` term correctly?

Answer 18

So it'd be the one of the answers with 7 and 2 in it?

Answer 19

yes.

Answer 20

So with the big numbers, I multiply that one?

Answer 21

What big numbers?
I'm not sure what you're talking about.

If you look at the 10th row of Pascal's Triangle,
you'll see that the numbers go like this

1 9 36 84 126 126 84 36 9 1

These are the coefficients for our expansion ^

Answer 22

Oh. Okay. So it'd be 36, then.

Answer 23

So it'd be D!

Answer 24

yes, good job.

sorry the site keeps crashing on me.

Answer 25

Thank you. I'll close this one now.