# more binomials

• Oct 2nd 2008, 02:16 PM
oxrigby
more binomials
I cannot figure out how to simplify binomials. (i) i can do (ii) i cant

(i) write down the general term in the expansion of $(1+x)^n$

(ii) use the term $(1+x)^m(1+x)^n=(1+x)^(n+m)$

to prove that: (this is binomials btw)

mCr+mCr-1.nC1+mCr-2.nC2+....+nCr=n+mCr where . is a multiple sign
• Oct 2nd 2008, 08:17 PM
mr fantastic
Quote:

Originally Posted by oxrigby
I cannot figure out how to simplify binomials. (i) i can do (ii) i cant

(i) write down the general term in the expansion of $(1+x)^n$

(ii) use the term $(1+x)^m(1+x)^n=(1+x)^(n+m)$

to prove that: (this is binomials btw)

mCr+mCr-1.nC1+mCr-2.nC2+....+nCr=n+mCr where . is a multiple sign

$(1 + x)^{n+m} = 1 + ^{n+m}C_1 x + \, .... \, + ^{n+m}C_r x^r + \, .... \, + x^{n+m}$ .... (1)

$(1 + x)^n (1 + x)^m$ $= \left( 1 + ^nC_1 x + \, .... \, + ^nC_s x^s + \, .... \, + x^n \right) \left( 1 + ^mC_1 x + \, .... \, + ^mC_r x^r + \, .... \, + x^m \right)$ .... (2)

Partially expand (2) to get the coefficient of $x^r$:

$^mC_r + ^mC_{r-1} ^nC_1 + \, ....$

Compare this with the coefficient of $x^r$ given in (1).
• Oct 3rd 2008, 12:23 AM
oxrigby
how do you partially expand this? D o you just take into account the first three terms in each bracket i.e 1,nC1x,nCrx^r multiply them together,equate them to (1+x)^n+m. Im also confused by the notation of r and s shouldn't it just be one of them''..
• Oct 3rd 2008, 01:32 AM
mr fantastic
Quote:

Originally Posted by oxrigby
how do you partially expand this? D o you just take into account the first three terms in each bracket i.e 1,nC1x,nCrx^r multiply them together,equate them to (1+x)^n+m. Im also confused by the notation of r and s shouldn't it just be one of them''..

Multiply out the terms that you can see will give you a coefficient of x^r. You might want to include more terms in each of the brackets than I did to help you see this.