An ice cream parlor has n toppings.
(a) How many i-topping ice cream cones are possible, where i = 0, . . . , n?
(b) Considering that an ice cream cone can have anywhere from 0 to n toppings, derive a formula for the total number of ice cream cones possible.
(c) We can also count the number of possible ice cream cones by considering each topping to be an experiment with two outcomes: on or off. Use the GBPC to determine the number of ice cream cones possible in this way.
(d) Equate your answers from parts (b) and (c) to derive an interesting identity involving binomial coefficients. (This identity can also be derived by setting x = y = 1 in the binomial theorem and has a neat implication for the sums of rows of Pascal’s triangle.)
I'm confused on the entirety of this problem, anything helps! Thank you!
For (a) I need a formula for how many possible i-toppings ice cream cones there are. We have to assume we can't double up on toppings. I have $\frac{n!}{2x}$. I don't think this is correct.
For (b) Would it be i=x, $x \in \mathbb{Z}$. Therefore the outcomes goes from 0 to n?
For (c) and (d) I have no attempt.