Question

The tenth term of (x+1)^20 is 167,960x^11. Write the twelfth term.( hint: The eleventh term is the middle term)

Answer 1

@amistre64 @SolomonZelman @yrelhan4

Answer 2

\[\left( x+a \right)^n ,Tr+1=nCr ~x ^{n-r}~a ^{r}\]
put n=20,r+1=12,r=12-1=11
a=1

Answer 3

\[nCr=20C11=20C(20-11)=20C9\]

Answer 4

I'm supposed to use Pascal's Triangle to figure this out... but i don't know how to apply it to this problem.

Answer 5

\[\frac{ 20*19*18*17*16*15*14*13*12 }{ 9*8*7*6*5*4*3*2*1 }=?\]

Answer 6

how did you get that? this is confusing :/

Answer 7

\[x ^{20-11}=x^9,a ^{11}=1^{11}=1\]
so term is 197960 x^9

Answer 8

in the previous solution i have used binomial theorem.

Term is \[x ^{11-2}=x^9\]
power of x goes on decreasing.

Answer 9

Hi, are you supposed to use the Pascal's Triangle?

Answer 10

yes, that's what they want me to usee.

Answer 11

If so, refer the 20th row. According to the hint, the eleventh term is the middle term, so you're looking for the number in the middle number in that row.

Answer 12

how do I find the 20th row?

Answer 13

It's really not viable to see Pascal's here. It's hard to write everything down to 20 rows. :P

Answer 14

Do you know where the coefficients in Pascal's Triangle come from?

Answer 15

(That's exactly what surjithayer showed you.)

Answer 16

each of the numbers is the sum of the two numbers above it?

Answer 17

Here are all the terms
http://www.wolframalpha.com/input/?i=expand+%28x%2B1%29^20

Answer 18

Yes, but I'm talking about another way.

Do you know about permutations and combinations?

Answer 19

no, I don't.

Answer 20

Hmm, then you'll have to refer to a huge version of Pascal's. Let me search for it.

Answer 21

yeah I'm learning permutations pretty much at the end of my algebra 2 textbook.

Answer 22

Here, I found the twentieth row.

{1, 20, 190, 1140, 4845, 15504, 38760, 77520, 125970, 167960, 184756,
167960, 125970, 77520, 38760, 15504, 4845, 1140, 190, 20, 1}

Find the eleventh number in that.

Answer 23

@eliassaab They haven't done combinatorics yet. :(

Answer 24

Oh wait a second, they need to find the twelfth term.

Answer 25

Then the twelfth number in that list of numbers is the coeff.

Answer 26

okay now how would i know what power x would be?

Answer 27

oh I know how, it starts out to the twentieth power and then gets smaller.

Answer 28

Polynomials are written from highest power of x to the lowest and so I would say x^9.

Answer 29

so yeah x^9.

Answer 30

Let's try to give you a simple example.\[(x+1)^2 = x^2 + 2x + 1\]The first term has power \(n\), the second has power \(n-1\) ... the \(k\)th term has power \(n - k+1\).

Answer 31

thanks everyone for the help... thanks for the twentieth row of Pascal's @ParthKohli

Answer 32

Hence, the twelfth term has power \(20 - 12 + 1 = 9\).

Answer 33

No problem!

Answer 34

and I just read on in my book and this was the next section of it!

Answer 35

Ah, cool.