Sum of following binomial series :

Question

I need to solve this binomial summation but cant seem to get it using binomial identities I learnt in school and college first-year:

$$S=\sum_{i=q}^{p-q}{\binom{i}{q}}{\binom{n-i}{p-q}}$$ p,q,n are positive integers such that $q\leq p\leq n$

Am I missing out something naive and clever?

sorry for confusing you with this thing actually the summation is to be done for all listed values of i and q need not be less than p-q — Shagun Sodhani, Feb 02 '13 at 13:59
Do you mean that the sum is actually $\sum_{i=p-q}^q$ if $p-q<q$? — Julien, Feb 02 '13 at 14:01
I meant that the sum is to be calucated for all values of i in range q to p-q both inclusive and I wrote it this way as I dont know before hand if p-q less tha q or not. — Shagun Sodhani, Feb 02 '13 at 14:03
If the range runs all the way up to $n$ then the answer, together with a few proofs, is given here. — Mike Spivey, Feb 02 '13 at 20:37
The summation is truncated independently of $n$, while it should probably be over all $i$ that would give a nonzero contribution. It is unlikely that there exists a formula for this. — Marc van Leeuwen, Apr 22 '13 at 14:00

score 1 · Answer 1 · answered Feb 02 '13 at 14:06

1

I guess it is $nCp$. Explanation : In the process of selecting p students from a group of n students ,You divide the group of n students into two groups. One having i students and other having n-i students. Now you select q students from the i students and rest p-q sudents from the rest students.As i would run from q to p-q(considering q<=p-q) and you sum them you will get the no. of ways in which you can select p students from n students.

answered Feb 02 '13 at 14:06

Abhra Abir Kundu

8,150

For $n=4,p=2,q=1$ you get $\binom11\binom31=1\times3=3\neq6=\binom42$ – Marc van Leeuwen Apr 22 '13 at 13:59

Sum of following binomial series :

1 Answers1