# Homework Help: A binomial identity

1. Apr 25, 2009

### chaotixmonjuish

$$\sumk=0n\binom{n}{k}2=\binom{2n}{n}$$

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

$$(n-k)\binom{n}{k}=n\binom{n-1}{k}$$
Right Side: Suppose you create a committe from $$\binom{n}{k}$$, 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.

Last edited: Apr 26, 2009
2. Apr 26, 2009

### tiny-tim

Hi chaotixmonjuish !

(try using the X2 and X2 tags just above the Reply box )
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.

3. Apr 26, 2009

### chaotixmonjuish

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. Apr 26, 2009

### tiny-tim

uhhh?

the RHS is the same number, no matter how many men (or women) there are.

5. Apr 26, 2009

### chaotixmonjuish

Uh oh, ha ha, now i'm confused....I feel like this binomial identiy has some really easy RHS.

6. Apr 26, 2009

### chaotixmonjuish

Does it just count the number of ways to form a committee size of n from 2n people?

7. Apr 26, 2009

### tiny-tim

Yup!

Now … pretend the 2n people are n men and n women

8. Apr 26, 2009

### chaotixmonjuish

Okay, so does it still mean n people regardless of gender?

9. Apr 26, 2009

### tiny-tim

Yes … the RHS is still the same …

we wouldn't muck around with that!