1. ## 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
Thanks

2. 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
Thanks
This is a special case of Pascal's rule.

3. hi again the following link takes you to the question http://i223.photobucket.com/albums/d...f/HPIM0534.jpg
thanks

4. Many thanks for the Pascal's Rule. Perfect.

5. 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.

6. 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

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

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

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

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

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

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

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

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