Expressing the binomial coefficients

[vanitymdl]

Homework Statement

Expressing the binomial coefficients in terms of factorials and simplifying algebraically, show that
(n over r) = (n-r+1)/r (n over r-1);

Homework Equations

The Attempt at a Solution

I honestly don't even know how to come about this problem...I really need help in this subject. Even how to start it would be great.

 

[chiro]

Hey vanitymdl and welcome to the forums.

What is the definition of nCr? Try using this definition and then simplify as much as you can. If you get stuck show us what you have done so you can get suggestions.

I'll get you started by saying that nCr = n! / [r! * (n - r)!] where x! is x factorial (i.e. you multiply all numbers from 1 to x and returns the result for x!). We also define 0! to be 1 and don't consider negative factorials to exist.

 

[vanitymdl]

I guess that part that is confusing me is how can I multiply (n-r+1)/r (n over r-1)?

 

[chiro]

I guess that part that is confusing me is how can I multiply (n-r+1)/r (n over r-1)?

By this do you mean [(n-r+1)/r] * nC(r-1) or (n-r+1) / [r * nC(r-1)] (or something else)?

 

[vanitymdl]

By this do you mean [(n-r+1)/r] * nC(r-1) or (n-r+1) / [r * nC(r-1)] (or something else)?

I mean, [(n-r+1)/r] * nC(r-1). So how would I come about that?

 

[chiro]

Since nC(r-1) = n! / [(r-1)! * (n - r + 1)!] then you have

[n-r+1]/r * nC(r-1) = n! * (n-r+1)/[r*(r-1)! * (n-r+1)!]

Now r*(r-1)! = r! and (n-r+1)/(n-r+1)! = 1/(n-r)! (you can just expand the factorial).

Using these hints, can you simplify further?

 

[vanitymdl]

And Yes you can... so the [n-r+1]/r * nC(r-1) will simplify to n!/r!(n-r)!

So this will lead me to the nCr which is n!/r!(n-r)!