Question

What does it mean?

Answer 1

Hey

Answer 2

hello again

Answer 3

What's the r'th term in the expansion of (a+b)^n ?

Answer 4

It is given by nCr*a^(n-r)*b^r . Right?

Answer 5

Yes, what's the r'th term in the expansion of (1+x)^n ?

Answer 6

It would be nCn*1^(n-n)*x^n ( nCn and 1^0 is 1 so it would be x^n).

Answer 7

Careful, I'm asking about the r'th term

Answer 8

Oops! nCr*x^r

Answer 9

Yes, r'th term in the expansion of (1+x)^n is nCr*x^r.

What must we replace "r" by to see x^(n-2) ?

Answer 10

n-2

Answer 11

Yes, what's the "n-2" th term in the expansion of (1+x)^n ?

Answer 12

nC(n-2)*x^(n-2)

Answer 13

Right, so the coefficient of x^(n-2), according to our formula of r'th term is nC(n-2)

Answer 14

Oh sorry, the exponent is 2n, not n. Let's start over...

Answer 15

then It would be 2nC(n-2)x^(n-2)

Answer 16

Perfect!

r'th term in the expansion of (1+x)^(2n) is (2n)Cr x^r

we get x^(n-2) term if we replace r by n-2 above.

Answer 17

therefore we want to find the n-2 th term in the expansion of (1+x)^(2n) :

(2n)C(n-2)*x^(n-2)

Answer 18

With me so far ?

Answer 19

Yes

Answer 20

simple expand (2n)C(n-2) and we're done

Answer 21

use the formula nCr = n!/[(n-r)! r!]

Answer 22

Oh! But in book they have expanded 2nC(n+2).

Answer 23

it's the same thing

nCr = nC(n-r)

Answer 24

Ahh right, it must be some printing mistake. Replace 2nC(n+2) by 2nC(n-2) and everything should make sense...

Answer 25

Hey! Go at brainly.com! You'll get instant answers!

Answer 26

ncr = nc(n-r)

but we don't want to use it here as rth term and n-r th terms wil have different degrees.

Answer 27

Also, why did they focused in x^(n-2)?...
They could have chosen x^(n-1).

Answer 28

Good question. This will be answered after we look at the right hand side of the equation iii.

Your goal here is to find an expression for C0C2 + C1C3 + C2C4 + ...., right ?

Answer 29

what is the full question?

Answer 30

Got it!! Thanks everyone..

Answer 31

You will be using this trick a lot in "generating functions" which you may be messing with soon. You get to find the coefficient of a certain term like, x^r, which represents number of ways of doing a particular thing. So it is okay if you don't get 100% of the problem now. You're gonna see more of these later and you will understand them after using this trick in different situations.