Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Possible Duplicate:
How to find a closed formula for the given summation

I am looking for a fast/best approach to compute $$\sum_{i=0}^{n} \binom{19}{i} \times \binom{7}{n-i}$$

For example,if $n=4$ the answer is $\binom{26}{4}$.Please explain your appraoch.

ADDED: After some experimentation with different numbers I think in general,$\sum_{i=0}^{n} \binom{p}{i} \times \binom{q}{n-i} = \binom{p+q}{n}$ holds.Does this result correct? If yes how could we proof this result.Any ideas?

share|improve this question

marked as duplicate by Zev Chonoles Sep 5 '11 at 20:03

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

2  
Guess the rule then prove it. –  Yuval Filmus Sep 5 '11 at 19:33
4  
    
@ Zev Chonoles:Thanks for the link :-) –  Quixotic Sep 5 '11 at 19:40
    
@Zev Chonoles:You may delete this question,I think the other question has already provided me with sufficient information that one could possible want to know about this problem.Thanks. –  Quixotic Sep 5 '11 at 20:02
    
@FoolForMath: I don't think there's a need to delete it, I will just close as duplicate. –  Zev Chonoles Sep 5 '11 at 20:03

1 Answer 1

up vote 3 down vote accepted

It comes very quickly.

This is just a way of finding out how many ways there are to choose n things from 26 objects. So for any n, the answer will be $26 \choose n$. How? Say we divide the 26 things into 2 groups, call 19 of them 'red' and 7 'blue.' Then we sum over the possibilities of taking 0 blue, n red or 1 blue, n-1 red, ... , n blue, 0 red. That is exactly your sum, and that is why it's just $26 \choose n$.

share|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.