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## #1 2010-10-18 02:30:36

1a2b3c2212
Member
Registered: 2009-04-04
Posts: 419

### Binomial Theorem

How do you simplify

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## #2 2010-10-18 02:36:48

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Binomial Theorem

19CK is the short way of saying:

And you can write 19C(k+1) in a similar way.

Therefore, the fraction in the question is:

When you also expand out all the factorials (eg. k! = 1x2x3x4x5x...xk), then a lot of things can cancel out and leave you with something relatively simple.

Why did the vector cross the road?
It wanted to be normal.

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## #3 2010-10-18 03:12:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Binomial Theorem

Hi 1a2b3c2212;

Divide top and bottom by 19!

Make use of the identity:

So 2) becomes:

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #4 2010-10-19 00:06:59

1a2b3c2212
Member
Registered: 2009-04-04
Posts: 419

Is this correct?

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## #5 2010-10-19 00:34:37

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Binomial Theorem

Hi 1a2b3c2212;

To prove a conjecture no amount of numerical evidence will suffice, but to disprove one you only need a single counter example.
So numerical experimentation ( plugging in numbers ) can be a powerful technique when trying to disprove something.

No it is not true, try the triplet ( n = 10 , k = 3 , r = 2 ) or an easier one to compute ( n = 7, k = 5 , r = 2 ).

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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