Deriving the Formula for Combinations with Repetitions

[Seda]

I need to know a way to derive the formula when doing combinations with repetitions.

THe formula is basically adapted from C(n;k) to

(n+k-1)!/(k!(n-1)!)
How is thing monster dervied?

i don't know if this will help, but the homework problem itself is:

How many distinguishable fruit baskets with 7 items can be created using apples, oranges, and pears?
I've used the formula to get an answer of 36. However, we haven't learned that formula in class, just the basic C(n,k) formula. Thats why I need to derive what I haven't learned in class from what I have learned.I've looked at this thing for like 30 minutes and I am like nowhere. The n+k part of the numerator I think i get because its adding what we take away back to the set we are choosing elements from. I don't get the -1 though. or the rest.

Help!

 

[Dick]

I think you really ought to state what problem that formula is supposed to solve.

 

[Seda]

See Next

 

[Seda]

SOrry for the extra posts, everythings in the first one now. Sorry.

 

[Dick]

That helps a lot. So n=7 and k=3. You want to divide a set of 7 objects into 3 groups. Think of 7 books on the shelf. You want to divide them into 3 groups. You can do this by inserting 2 dividers into the row. They are allowed to fall at the ends, in that case the missing group has zero items. So you have 9 items on the shelf, 7 books and 2 dividers. The total number of ways to do this is to take 9 items and select any 2 to be the dividers. That's C(9,2). Or in terms of n and k, C(n+k-1,k-1). I'm not sure this is totally clear, but I tried.

 

[Seda]

I was under the impression the n=3 and k=7. I have 3 fruits and I'm making groups of 7.

This is possible because I can pick any fruit more than once.

 

[Dick]

If you want to interchange the meaning of n and k, that's fine. The argument still works. It's still C(9,2).