An ice cream parlor has 6 toppings to choose from (hot fudge, sprinkles, caramel, cherries, blueberry sauce, and oreos.
How many ways are there to make a sundae with two different toppings?
So for this question I believe its a combination and the number of possibilities would be 6C2.
How many ways are there to make a sundae with two toppings if double toppings are allowed?
So since each topping has the option of being doubled would it be 12C2? I'm not really sure how to go about this.
What would happen if there were three toppings with triple toppings allowed?
Best Answer
In the two-toppings with repetition case, the answer is $\binom{7}{2}$. You want the number of mappings from 2 balls to a 5 boxes, up to permutations of the balls. The general formula for $n$ balls and $x$ boxes is $\binom{x+n-1}{n}$
This can be counted with stars and bars: You have 7 positions $*******$, and are choosing 2 of them to be the "stars"/balls, and the remaining 5 delimit the boxes. One such placement is $||*||*|$. If a star appears before the first bar, the first topping is used. If a star appears between the first and second bars, the second topping is used.... If a star appears after all of the bars, the 6th topping is used.
If 2 stars appear in any of the gaps between bars, that topping is used twice.
Three toppings with triple toppings allowed can be counted in the same way: $n=3$ and $x=6$, so your number of choices is $\binom{6+3-1}{3}=\binom{8}{3}$.