Binomial Q..

Question

Binomial Q..

priyar · Answer

$$\left(\begin{matrix}1000 \ 50\end{matrix}\right) +\left(\begin{matrix}999 \ 49\end{matrix}\right)+\left(\begin{matrix}998 \ 48\end{matrix}\right)+.......+\left(\begin{matrix}949 \ 0\end{matrix}\right)$$=?

priyar · Answer

@ganeshie8

priyar · Answer

@ParthKohli

parthkohli · Answer

Hockey stick, eh? :P

priyar · Answer

no.. this is what i got as the exp. while solving a problem..

priyar · Answer

and thats not the answer..

ganeshie8 · Answer

There seems to be an issue here

1000 to 949 are 52 numbers
50 to 0 are 51 numbers

parthkohli · Answer

Oh haha, good observation.

priyar · Answer

oops! thanks! its 950

parthkohli · Answer

Hockey Stick Identity http://www.artofproblemsolving.com/wiki/index.php/Combinatorial_identity

priyar · Answer

pls tell me how to find the sum..

ganeshie8 · Answer

The answer is then simply $\dbinom{1001}{50}$

priyar · Answer

but how?

ganeshie8 · Answer

First look at what the identity says
we can bother about proving afterwards

ganeshie8 · Answer

you must be familiar with pascal's triangle and how it relates to the binomial coefficients, right ?

priyar · Answer

yes..

ganeshie8 · Answer

Good. Add the blue numbers, what do you get ?

|dw:1456992132781:dw|

priyar · Answer

oh ok.. i am reading from the link...pls wait..

ganeshie8 · Answer

take your time

priyar · Answer

its a bit confusing.. can you write the general formula and the one for the above exp. to make things clearer..

ganeshie8 · Answer

Forget about the link. The identity is easy if you interpret it using the pascal's triangle

priyar · Answer

yes.. i understood the actual meaning of the identity...that is if we add all the no's in a diagonal in the pascal's triangle..the sum will be equal to the number directly perpendicular to it.. and this shape looks like a hockey stick. right?

priyar · Answer

But i didn't understand the mathematical formulation of it..

ganeshie8 · Answer

$$\sum\limits_{i=r}^n \dbinom{i}{i-r} =  \dbinom{n+1}{n-r}$$

plugin $n=1000,~~r = 950$

priyar · Answer

why i=r wouldn't the "i-r" term be zero always?

priyar · Answer

@ParthKohli ?

ganeshie8 · Answer

Here $i$ is the only variable

$n$ and $r$ are fixed

priyar · Answer

ok now i  got it! thanks @ganeshie8 !