1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Binomial Theorem

  1. Dec 26, 2004 #1
    Binomial Theorem......


    I need to know about binomial theorem....

    eg. how to in general expand (a+b)^n

    I dont understand the combinations / permutations....?


  2. jcsd
  3. Dec 26, 2004 #2
  4. Dec 26, 2004 #3
    And this thread.
  5. Dec 26, 2004 #4
    Another expansion can be obtained using the Taylor Function.


    So in your case,

    [tex]f(x) = (a+b)^x[/tex]

    P.S. If I remember correctly, this only applies for real numbers.
  6. Dec 27, 2004 #5
    what i dont understand is the combinations and permutations........
    when used in binomial theorem ?

    what does it mean n choose x ?

  7. Dec 27, 2004 #6
    the thing I am used to is

    [tex] ^n C _r [/tex]

    i just use the one in my calculator, yours should have too. in fact, my teacher taught us the binomial theorem before permuations and combinations !
  8. Dec 27, 2004 #7
    Lets just expand some simple brackets first:
    [tex](a+b)^0 = 1[/tex]
    [tex](a+b)^1 = a+b[/tex]
    [tex](a+b)^2 = (a+b)(a+b) = a^2+ab+ba+b^2 = a^2+2ab+b^2[/tex]
    [tex](a+b)^3 = (a+b)(a+b)(a+b) = (a+b)(a^2+2ab+b^2) =

    a^3+2a^2b+ab^2+a^2b+2ab^2+b^3 = a^3+3a^2b+3ab^2+b^3[/tex]
    As you can see I have expanded the brackets the normal (well normal to me) way to do so. The problem that the binomial solves is when you have [tex](a+b)^1^5[/tex]or something in that nature.

    Also it is possible to see that there is a pattern to the coefficents (the numbers before the [tex]ab[/tex] or the [tex]ab^2[/tex]). The pattern, if wirtten out, is a pascal triangle. The top row has a single 1 then row two as two 1's (either side of the original one) and so on. This like will show you what I mean and then look back at the brackets above. The coefficents match (http://mathworld.wolfram.com/PascalsTriangle.html).

    Lets take the [tex](a+b)^2[/tex] and work it out using the binomial theorem.
    The first part is [tex]a^2[/tex]. We know this without any working out that occurs on paper but using the binomial it would be:
    [tex](a)^2[/tex] which gives [tex]a^2[/tex].
    The [tex]2ab[/tex] part is denoted by [tex]2(a^1b^1)[/tex] which gives [tex]2ab[/tex].
    The last part of this is similar to the first but using b instead of a.

    The pattern is: [tex](a+b)^2 = ^2 C _0(a^2(b)^0) + ^2 C _1(a^1(b)^1) + ^2 C _2(a^0(b)^2) = 1(a^2(1)) + 2(a(b)) + 1(1(b^2)) = a^2+2ab+b^2[/tex]
    As you can see, the indices add up to the original indice.

    A harder one just to show what the binomial does:
    [tex](a+b)^6 = ^6 C _0(a^6(b)^0) + ^6 C _1(a^5(b)^1) + ^6 C _2(a^4(b)^2) + ^6 C _3(a^3(b)^3) + ^6 C _4(a^2(b)^4) + ^6 C _5(a^1(b)^5) + ^6 C _6(a^0(b)^6)[/tex]
    [tex]= 1(a^6(1)) + 6(a^5(b)) + 15(a^4(b)^2) + 20(a^3(b)^3) + 15(a^2(b)^4) + 6(a^1(b)^5) + 1(a^0(b)^6)[/tex]
    [tex]= a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6[/tex]

    To finish the reason that I put the [tex]b[/tex] term in brackets without the indice is because if you get [tex](a-b)^n[/tex] then the part of the equation could be negative (e.g. [tex]20(a^3(-b)^3) = -20a^3b^3[/tex]).

    I hope this helps.

    The Bob (2004 ©)
    Last edited: Dec 28, 2004
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook