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July 24th, 2017, 08:42 AM   #1
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Combinations with Variable r

Boy there are lots of different questions about combinations on this site, but none seem to match mine.

How do I figure number of combinations of n items - no repeats - but there is no limit on r, and order doesn't matter?

In other words, think of a pizza menu.

If there are two potential toppings, there are three options:
  1. Pepperoni (a)
  2. Sausage (b)
  3. Pepperoni and Sausage (ab)

If there are three toppings, I get seven options (I think)
  1. Pepperoni (a)
  2. Sausage (b)
  3. Mushrooms (c)
  4. Pep and Sausage (ab)
  5. Pep and Shrooms (ac)
  6. Sausage and Shrooms (bc)
  7. The Works (abc)

Four toppings = 15 options (I think)
  • a, b, c, d
  • ab, ac, ad, bc, bd, cd
  • abc, abd, acd, bcd
  • abcd

Five toppings = 30 options (I think...)
  • a, b, c, d, e
  • ab, ac, ad, ae, bc, bd, be, cd, ce, de
  • abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde
  • abcd, abce, abde, bcde
  • abcde

I can't find a pattern!
15 = 4!-(3^2) (i.e., n!-((n-1)^2)))
30 = 5!/(n-1)

Ugh!

How am I not looking at this right?
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July 24th, 2017, 09:10 AM   #2
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You missed acde. If you include "no toppings", the results become 4, 8, 16, 32, etc.
Thanks from Alonzo Mosley
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July 24th, 2017, 09:19 AM   #3
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Arghhh!!! I DID!!! ARGGGHH!!..

So,

(2^n)-1

Argh. Yes. Thanks. Wow, that hurt.
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