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Math Help - problems with (!) function

  1. #1
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    problems with (!) function

    Im not sure what ! is called but my question is how is

    (x+1)! / x! = (x+1)

    also is there a formula to solve

    (x+1)^a where a equals a # (for example 18)
    Last edited by Reminisce; July 20th 2010 at 11:35 PM.
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  2. #2
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    Quote Originally Posted by Reminisce View Post
    Im not sure what ! is called but my question is how is

    (x+1)! / x! = (x+1)
    Dear Reminisce,

    \frac{(x+1)!}{x!}=\frac{(x+1) \times (x) \times (x-1) \times ...... \times 2 \times 1}{(x) \times (x-1) ..... \times 2 \times 1} =x+1

    Did you understand now?
    Last edited by mr fantastic; July 21st 2010 at 01:22 PM. Reason: Fixed latex.
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  3. #3
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by Reminisce View Post
    Im not sure what ! is called but my question is how is

    (x+1)! / x! = (x+1)

    5!=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1
    or
    5!=5 \cdot4 !
    n!=n \cdot (n-1) \cdot (n-1) \cdot (n-2) \cdot \cdot \cdot (n-m)!
    so u have...
    \frac{(x+1)!}{x!}=\frac{(x+1) \cdot x!}{x!}=(x+1)
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  4. #4
    Member Mathelogician's Avatar
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    Infact it's called Factorial.
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  5. #5
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Reminisce View Post
    Im not sure what ! is called but my question is how is

    (x+1)! / x! = (x+1)

    also is there a formula to solve

    (x+1)^a where a equals a # (for example 18)
    In general is x is a real number with x>-1 [so that it isn't neccessarly an integer...] the factorial function [or 'Pi function' according to Gauss interpretation...] is defined as a definite integral that I don't report because the details are a little complex. If You want more details about it see...

    Gamma function - Wikipedia, the free encyclopedia

    In particular the 'facrorial function' satisfies the relation...

    x! = x\ (x-1)! (1)

    Till now we are 'all right'... a little less clear for Me is the second part of Your question, regarding the expression (x+1)^{a}... could You supply more details please?...

    Kind regards

    \chi \sigma
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  6. #6
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    Thank you very much
    as for the second part i was wondering if there is a formula to FOIL anything bigger then 3
    example
    (x+1)^18
    because it would be a hassle to write (x+1) eighteen times and FOILing it like that
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  7. #7
    Senior Member yeKciM's Avatar
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    u can use : (sorry i do'nt know how this LaTex work just jet)

    _____1_____
    ___1_2_1___
    __1_3_3_1__
    _1_4_6_4_1_
    ...................

    for let's say 10th... it would be... 1_10_45_120_210_252_210_120_45_10_1

    and u'll have
    (x+1)^{10}=x^{10}+10x^{9}+45x^{8}+120x^{7}+210x^{6  }+252x^{5}+210x^{4}+120x^{3}+45x^{2}+10x^{1}+1


    and so on.... but if u need to know let's sey n-th member of that u can use binomial template (or something... I'm bad with English... more or less)
    Last edited by yeKciM; July 21st 2010 at 06:00 AM.
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  8. #8
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    I'm still a bit confused. Like how did you get the values 1_10_45_120....
    from using the pyramid
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  9. #9
    Senior Member yeKciM's Avatar
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    well... when u do pyramid ... see that at each side are ones... but numbers inside are sum od two numbers from above...
    look at 4... firs is one ... second is (above 1+3 =4) 4 .... 3rd is (above 3+3 =6) 6 ... 4th is (above 3+1=4) 4 and 5th is 1
    and so on...

    or like this :

    (a+b)^n = a^n + \binom {n}{1}a^{n-1} b + \binom {n}{2}a^{n-2}b^2 + \cdot \cdot \cdot + \binom{n}{n-1}a b^{n-1}+b^n

    or if u need let's say (k+1)th member u'll use this one :

    T_{(k+1)}=\binom {n}{k}a^{n-k}b^k that's one if u need (k+1)th member from begining ...

    T_{(k+1)}=\binom {n}{k}a^{k}b^{n-k} that's one if u need (k+1)th member from end ...
    Last edited by yeKciM; July 21st 2010 at 07:30 AM.
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  10. #10
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Reminisce View Post
    I'm still a bit confused. Like how did you get the values 1_10_45_120....
    from using the pyramid
    Pascal's triangle - Wikipedia, the free encyclopedia

    Also for doing binomial coefficients you might find this useful

    Combinations and Permutations Calculator

    Choose No - No from dropdown menus, then for example n = 18 and r = 5 will give you the coefficient in front of the x^5 term of (x+1)^18 expanded. See

    Binomial coefficient - Wikipedia, the free encyclopedia
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  11. #11
    MHF Contributor Unknown008's Avatar
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    If you want to expand (x+1)^a, then you get:

    (x+1)^a = x^a + \binom{a}{1}(x^{a-1})(1^1) + \binom{a}{2}(x^{a-2})(1^2) + ... + \binom{a}{a}(x^{a-a})(1^a)

    Which is equivalent to:

    (x+1)^a = x^a + \binom{a}{1}(x^{a-1}) + \binom{a}{2}(x^{a-2}) + ... + 1

    As 1 to the power of anything will give 1.

    Now, \binom{a}{1} = \frac{a}{1!}

    \binom{a}{2} = \frac{a(a-1)}{2!}

    Or, \binom{a}{n} = \frac{a!}{(a-n)!(n!)}

    If that is what you were looking for...

    EDIT: Oops. I didn't see your post yekcim =S
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  12. #12
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by Unknown008 View Post

    EDIT: Oops. I didn't see your post yekcim =S
    lol it's my fault... it take me too long time to see how to write \binom hehehehe so i didn't see urs

    and to make a note that :

    \binom {n}{0} = 1

    \binom {0}{0} = 1

    \binom {n}{n} = 1

    \binom {n}{k} = \binom {n}{n-k}
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  13. #13
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    o.O thanks but all those equations confused me but i did however found out how to do them Using Pascal's Triangle, thanks!

    This is the way i figured it out:

    (a+b)^n

    X(a^nb^0)

    X = pascal's number
    n = n but as we move to the right we subtract 1
    0 = 0 but as we move to the right we add 1
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  14. #14
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    Quote Originally Posted by chisigma View Post
    In general is x is a real number with x>-1 [so that it isn't neccessarly an integer...] the factorial function [or 'Pi function' according to Gauss interpretation...] is defined as a definite integral that I don't report because the details are a little complex. If You want more details about it see...
    No its not, it is a function defined on the natural numbers. There is a way of extending it as you suggest (to a larger domain as well), but they are different things, and what you suggest here is useless to the poster of a question in the pre-university section of MHF

    CB
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