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A binomial identity

  • #1
[tex]\sumk=0n\binom{n}{k}2=\binom{2n}{n}[/tex]


Could someone give me a hint as to how to start this. I'm not sure how to really interpret it.



[tex](n-k)\binom{n}{k}=n\binom{n-1}{k}[/tex]
Right Side: Suppose you create a committe from [tex] \binom{n}{k} [/tex], then to pick a leader who isn't in the committee but in the pool of people, we have n-k ways.

Left Side: Suppose you have n ways to pick a leader for a group. After selecting the leader, you have n-1 people left to pick a committee of size k.
 
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Answers and Replies

  • #2
tiny-tim
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Hi chaotixmonjuish ! :smile:

(try using the X2 and X2 tags just above the Reply box :wink:)
k=0n nCk2 = 2nCn

Could someone give me a hint as to how to start this. I'm not sure how to really interpret it.
The RHS is the number of ways of choosing n people from 2n people.

Hint: Suppose the 2n people are n men and n women. :wink:
 
  • #3
So would the right hand side be saying that suppose we had n men and n women, there are n ways to form a committee consisitng of both men and women.
 
  • #4
tiny-tim
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So would the right hand side be saying that suppose we had n men and n women, there are n ways to form a committee consisitng of both men and women.
uhhh? :confused:

the RHS is the same number, no matter how many men (or women) there are.
 
  • #5
Uh oh, ha ha, now i'm confused....I feel like this binomial identiy has some really easy RHS.
 
  • #6
Does it just count the number of ways to form a committee size of n from 2n people?
 
  • #7
tiny-tim
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Does it just count the number of ways to form a committee size of n from 2n people?
Yup! :biggrin:

Now … pretend the 2n people are n men and n women :wink:
 
  • #8
Okay, so does it still mean n people regardless of gender?
 
  • #9
tiny-tim
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Okay, so does it still mean n people regardless of gender?
Yes … the RHS is still the same …

we wouldn't muck around with that! :rolleyes:
 

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