Choosing toppings for pizza Problem: Ordering a "deluxe" pizza means you have four choices from 15 available toppings. How many combinations are possible if toppings cannot be repeated? If they can be repeated? Assume the order in which the toppings are selected does not matter.
If the toppings cannot be repeated, then we have $C_4^{15}$ choices.
I am having difficulty with the toppings can be repeated problem.  It feels like this isn't solvable with the stars and bars method because we only care about "4" of the "15" toppings and not all 15.
What I think: Use "14" bars to then have "15" categories, each responding to a possible topping.  Have 4 stars to represent choosing a topping from that category.  Then we should get $C^{18}_{4}$?

if they can be repeated, it can be stars and bars - you have 15 things to choose from, but you only want to choose 4 - suppose you chose only 4 onion
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is a basic representation of 4 going into 15 slots
4 stars and 14 bars
so it is $\binom{18}{4} $
if they can't be repeated, then it is 4 from 15 where there is one of each, without replacement and without order, so it is $\binom{15}{4}$
