• Support PF! Buy your school textbooks, materials and every day products Here!

Proving this binomial identity

  • #1
[tex]\sum_{m=k}^{n-k}\binom{m}{k}\binom{n-m}{k}=\binom{n+1}{2k+1}[/tex]


I'm not sure how to prove it, I understand the combinatorial proof..i.e. putting it to an example...but i can't derive one side and get the other.
 

Answers and Replies

  • #2
290
2
How do you understand the combinatorial proof but not know how to prove it?
 
  • #3
It was an example given in the book. I just don't see how they manipulated the binomial to move the stuff around.
 
  • #4
Suppose you had a bag full of n balls. Suppose out of the n balls you had m green ones. Would the right hand side be adding up the ways to count all the gree balls and non-green balls. I didn't even really understand this reasoning. This identity is really frustrating me.
 

Related Threads for: Proving this binomial identity

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
11
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
6
Views
785
Top