Binomial Theorem proof

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June 29th 2010, 08:37 AM
minicooper58

Binomial Theorem proof

Hi this question was in Advanced Higher Maths in the UK
Show that (n+1 , 3) - (n ,3 )=(n ,2)
They are written in column vector form ie n+1 at top and 3 at the bottom for example for the first. I think you have to use the n c r formula to prove the rhs
Please Help
Thanks
June 29th 2010, 08:44 AM
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Quote:

Originally Posted by minicooper58

Hi this question was in Advanced Higher Maths in the UK
Show that (n+1 , 3) - (n ,3 )=(n ,2)
They are written in column vector form ie n+1 at top and 3 at the bottom for example for the first. I think you have to use the n c r formula to prove the rhs
Please Help
Thanks

This is a special case of Pascal's rule.
June 29th 2010, 08:47 AM
minicooper58

hi again the following link takes you to the question http://i223.photobucket.com/albums/d...f/HPIM0534.jpg
thanks
June 29th 2010, 09:06 AM
minicooper58

Many thanks for the Pascal's Rule. Perfect.
July 5th 2010, 09:54 AM
minicooper58

Hi, I understand that Pascal's rule is involved but I'm not sure how to go about the proof. The link to the question no 5 is at the following link http://i223.photobucket.com/albums/d...f/HPIM0534.jpg. It just seems to easy to simplify and make r =3.
July 5th 2010, 11:04 AM
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Quote:

Originally Posted by minicooper58

It just seems to easy to simplify and make r =3.

Well if you don't want to use Pascal's rule, try this

$\displaystyle \binom{n+1}{3} - \binom{n}{3}$

$\displaystyle =\frac{(n+1)!}{3!(n-2)!} - \frac{n!}{3!(n-3)!}$

$\displaystyle =\frac{(n+1)!-n!(n-2)}{3!(n-2)!}$

$\displaystyle =\frac{(n+1)(n!)-(n!)(n-2)}{3!(n-2)!}$

$\displaystyle =\frac{(n!)(n+1-(n-2))}{3!(n-2)!}$

$\displaystyle =\frac{3n!}{3!(n-2)!}$

$\displaystyle =\frac{n!}{2!(n-2)!}$

$\displaystyle =\binom{n}{2}$