>>12248517how many ways can i choose r=5 indistinguishable objects from n=8?
assume objects are distinguishable.
if i picked all the objects (without replacement), i'd have 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = n! ways to pick objects out since i have 8 choices the first pick, 7 choices the second, etc.
but i'm picking out 5, so i have 8 * 7 * 6 * 5 * 4 choices
put this into factorial notation: 8 * 7 * 6 * 5 * 4 = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 / (3 * 2 * 1) = 8! / (8 - 5)! = n! / (n - r)!
i assumed things were distinguishable, but they aren't. i over counted by the number of possible permutations for each set of 5 objects. the number of permutations is 5 * 4 * 3 * 2 * 1 = 5! = r!
divide out the overcounting: n! / (r! (n - r)!)
define this quantity as (n, r) := n! / (r! (n - r)!)
let n = r + m - 1 (for some reason)
plug this into the formula and turn the crank: (r + m - 1, r) = (r + m - 1)! / (r! (r + m - 1 - r)!) = (r + m - 1)! / (r! (m - 1)!)
this is the formula in OPs post
