Question

Can someone help me come up w/ a way to remember the forumla for Combination w/ repetition?:
(n+r-1)!
----------
r! (n-1)!
I have managed to remember the other 3 but I'm having a hard time deriving how this one was created.

Answer 1

Ok, think about it this way.

Imagine that we are at Sees candy store. And we want to fill a box with 10 different peices of candy, and they sell 3 different types of candy. So we want to pick how many of each type we will put in the box. We can represent these arrangements with a sequence of 'stars and bars' where the stars are the pieces of candy, and the bars separate each different type of candy.

So for example, if we wanted to arrange them so we could have our selection arranged thusly:
**********||
Where our box has only one type of candy. or we can do something more balanced:
***|***|****
etc.

So essentially we are finding how many different ways we can arrange the stars and bars. There are k+n-1 different slots (k candy spaces and n-1 bars representing the divisions between types of candy). And from that we are seeing how many different ways we can place the n-1 different bars


And if you can remember that \[C(n,k) = \frac{n!}{k!(n-k)!}\]
Then plugging in for the repetition formula is easy:

So we have that combinations with repetitions are
\[C_r(n,k) = C(n+k-1, n-1) = \frac{(n+k-1)!}{(n-1)!(n+k-1-n+1)!} = \frac{(n+k-1)!}{(n-1)!k!} \]

Answer 2

brilliant! thank u

Answer 3

Certainly! It was good for me to refresh my memory on this subject anyway, since it's been a few months since I studied it.

Answer 4

no hes just a showoff

Answer 5

That too!

Answer 6

lol

Answer 7

i don't know who likes to show off more,
you or satellite?

Answer 8

hahaha, nevertheless u r greatly appreciated

Answer 9

i out "algebraed" satellite today! :)

Answer 10

Probably me. Satellite is in it for the medals. Whereas I prefer to find questions where I get to show off my elite LaTeX skills.

Answer 11

lol

Answer 12

I came here thinking I might get to throw out a few \(n \choose r\) deals, but alas it was not to be.

Answer 13

Though if I were really hardcore I suppose I could have done the stars/bars chart with an array..... hrm...

Answer 14

http://openstudy.com/users/myininaya#/users/myininaya/updates/4e1bab140b8bc22757481e7a
see he admitted to being out algebraed!

Answer 15

Nice! =)

Answer 16

i love to win

Answer 17

I rarely win. I console myself on being correct.

Answer 18

Except of course when I'm not

Answer 19

whatever
you are a 99% winner

Answer 20

I'm much to slow. But I'm often right even if I arrive at the correct answer much later than other people.

Answer 21

you need speed when competing against satellite and you