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Math Help - more binomials

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by oxrigby View Post
    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).
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  3. #3
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    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''..
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  4. #4
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    Quote Originally Posted by oxrigby View Post
    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.
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