1. ## Pascal's Rule

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
it for this question using the identity:

(n , r-1)+(n ,r)=(n+1, r) written as column vectors

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
it for this question using the identity:

(n , r-1)+(n ,r)=(n+1, r) written as column vectors

Thanks
I see you are unsatisfied by my responses in this thread. Why not just ask about what you didn't understand?

3. ## Pascal's Rule

Combinatorial proof:

Pascal's Rule states:

$\displaystyle {n\choose k} = {n-1\choose k} + {n-1 \choose k-1}$

Proof:

Let $\displaystyle x$ be an element of group of $\displaystyle n$ elements.
The number of combinations of $\displaystyle k$ elements from group of $\displaystyle n$ elements, which is not containing the element $\displaystyle x$ is give by: $\displaystyle {n-1\choose k}$, now, the number of combinations which containing element $\displaystyle x$ is: $\displaystyle {n-1 \choose k-1}$.

But, every combination is containing $\displaystyle x$ or not containing $\displaystyle x$. Hence, the total number$\displaystyle {n\choose k}$ of combinations of $\displaystyle k$ from $\displaystyle n$ elements is the sum of the above.

Q.E.D