Page 38 - Math SL HB Sem 1

-

Bhio\ il-iI f Hi:a)R F^,.1

( n n)
Note: C 0 1
t r) 0!(n - 01!

nl nx(n-I)t
l!(n-1)! 1l(n - )!
I

-
-
: n! n x (n l) x (n 2)l _ n x (n - 1)
[;) ", 2!(n - 2)r. 2!x(n -2)l ?l
nx\n*l)x(n-2)(n-3)t _ nx(n-1)(n-2)
(;) = .,
3!(r-3)1 3!x(n-3)! l!

BINOMIAL EXPANSION
!
J^rivri'r.rl incziilr. [*u tcrrrii.; in a fonn of (xty)israLsedtoapowcr oro - btnomrai
expansion
Consider ( x + l,) n. we will €let the expansion as below:

n - o; l
n=1; x +
_v
n:2; ,' * 2xy + y'

n-3, *t + lx2y + I xyr n y'
n:4, -*o ; 4xly + 6*'y2 .+ ,4 xr'l + y{
'fhe
and so fortl.r. partem will continue till the desired n . The process is boring and lons.

Therefore a technique called Binomial Thectrem is being introduced.
Binomial Theorem

Definition:

| tx
(-t+r') = .' . ,n'v + t1 ) * "-'" * [''] *.'" -r- I nt v
[;) 2 ) [3/ lr)

n
+ ... + xv nl +
nl

, n(n-1\ nl
2!
I n xn'v', n,here n e Zu

r=0