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    I was playing around with binomial expansion and came up with a nifty proof inspired by a question DFranklin helped me with.

    \binom {n} {r} \times \binom {n-r} {1} = (r+1) \times \binom {n} {r+1}

    Would this be useful anywhere?
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    (Original post by Louisb19)
    I was playing around with binomial expansion and came up with a nifty proof inspired by a question DFranklin helped me with.

    \binom {n} {r} \times \binom {n-r} {1} = (r+1) \times \binom {n} {r+1}

    Would this be useful anywhere?
    This looks pretty cool, I'll try proving it tomorrow morning, when I wake up.
    (did you prove it the way I have in my groggy head right now, i.e: factorial definition and then recombining into binomial coefficient?)

    I can't immediately see where it's useful or comes handy, and I do suspect that if you ever come across such an occasion where it might be a shortcut, you won't remember this result anyway.
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    (Original post by Zacken)
    This looks pretty cool, I'll try proving it tomorrow morning, when I wake up.
    (did you prove it the way I have in my groggy head right now, i.e: factorial definition and then recombining into binomial coefficient?)

    I can't immediately see where it's useful or comes handy, and I do suspect that if you ever come across such an occasion where it might be a shortcut, you won't remember this result anyway.
    Yeah I used the factorial definition.

    The idea that (n-r)! = (n-r)(n-(r+1))! is what was I mainly used to simplify it down. I don't really see a use for it however I think when you have a binomial in the form

     (1 + x + y)^n

    The term  x^a \times y will have the coefficient of  \binom {n} {a} \times \binom {n-a} {1}

    (this might be wrong, I can't remember the question exactly)
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    (Original post by Louisb19)
    I was playing around with binomial expansion and came up with a nifty proof inspired by a question DFranklin helped me with.

    \binom {n} {r} \times \binom {n-r} {1} = (r+1) \times \binom {n} {r+1}

    Would this be useful anywhere?
    Looks interesting. If you like this sort of stuff try Q7 on https://share.trin.cam.ac.uk/sites/p...ampletest1.pdf
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    (Original post by Louisb19)
    I was playing around with binomial expansion and came up with a nifty proof inspired by a question DFranklin helped me with.

    \binom {n} {r} \times \binom {n-r} {1} = (r+1) \times \binom {n} {r+1}

    Would this be useful anywhere?
    I'm not sure it's useful to write \binom{n-r}{1} on one side and r+1 on the other.

    The more common thing would be to write:

    \binom{n}{r+1} = \dfrac{n-r}{r+1} \binom{n}{r}.

    Which is occasionally useful, particularly if you want to see what the largest term in a (finite) binomial distribution is going to be. (Most commonly this comes up in questions asking what the most likely value for a random variable is going to be).
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    (Original post by Louisb19)
    I was playing around with binomial expansion and came up with a nifty proof inspired by a question DFranklin helped me with.

    \binom {n} {r} \times \binom {n-r} {1} = (r+1) \times \binom {n} {r+1}

    Would this be useful anywhere?
    I did the proof and I'm inclined to agree with DFranklin in that it's slightly useless to write it in the form you have, and it would be better served in the form he's given.

    I can't quite think of any applications where this might come into useful, but if I come across anything, I'll post back.
 
 
 
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