Learning combinations vs permutations

Just one of the things I'm learning. https://github.com/hchiam/learning

A reference: https://www.mathsisfun.com/combinatorics/combinations-permutations.html

Combinations are different from permutations. But did you know that there are actually different kinds of combinations and different kinds of permutations?

Number of: (n = options, r = choose)

1. permutations with repeats allowed: n^r
2. permutations without repeats (if r == n): n!
3. permutations without repeats (if r != n): n! / (n-r)!
   • intuition: example: 5 pick 2 = 5*4, not 5*4*3*2*1
   • intuition: you're picking the first 2, hence 5*4
   • intuition: n! / (n-r)! is just a fancy math way of getting that 5*4
   • intuition: divide by (unwanteds) = (n-r)! = 3*2*1 in the example
4. (combinations = permutations but order doesn't matter)
5. combinations without repeats: n! / (n-r)! / r!
   • intuition: (permutations) / (ways to order r)
   • intuition: (permutations) = n! / (n-r)!
   • intuition: (ways to order r) = r!
6. combinations with repeats allowed: (n-1 + r)! / (n-1)! / r!

To-do:

Understand why there's -1s in the equation for combinations with repeats allowed.

The rest of that last equation makes sense:

• n to be combined + r repeats allowed = (n + r)! permutations
• /n!r! to remove the ways to order n items and r items
• both of the above give you (n+r)! / n!r! "combinations"

So why the -1 on the top and bottom?