How to add this combinations? PLEASE HELP

Question

How to solve this?

$\dbinom{1000}{50} + \dbinom{999}{49} + \dbinom{998}{48} + \dbinom{997}{47} +\cdots+ \dbinom{951}{1} + \dbinom{950}{0}$

I was solving some problem which goes like this

Not knowing any other better method (if exists), I thought of adding the coefficients of x^50 from each term!

And reached the step as depicted above.

This is a diagonal on Pascal's triangle and there are nice identities. Have a search, see if you can find it. You might want to look up triangular numbers. https://math.stackexchange.com/questions/1413046/pascals-triangle-sum-of-nth-diagonal-row — it's a hire car baby, Mar 23 '18 at 08:26

score 4 · Accepted Answer · answered Mar 23 '18 at 07:45

4

Hint

The expression is same as $$\binom {1000}{950}+\binom {999}{950}+\binom {998}{950}\cdots +\binom {950}{950}$$

And now use the Hockey stick identity to get the answer as $\binom {1001}{951}$

answered Mar 23 '18 at 07:45

Rohan Shinde

score 3 · Answer 2 · answered Mar 23 '18 at 07:49

3

$$\binom nk=\binom{n-1}{k-1}+\binom{n-1}k$$

$$\iff\binom nk-\binom{n-1}k=\binom{n-1}{k-1}$$

$$\implies\binom{950+k}k-\binom{950+k-1}k=\binom{950+k-1}{k-1}$$

Set $k=1,\cdots,50$ and add

answered Mar 23 '18 at 07:49

lab bhattacharjee

score 3 · Answer 3 · answered Mar 23 '18 at 07:51

3

From Pascal's triangle, the idea behind is this

answered Mar 23 '18 at 07:51

user

3 Answers3