Question

pedro's school had an ice cream social to help raise money for a school trip. how many ways can ice cream be ordered with 2 different toppings ?

Answer 1

So, say order mattered. If we had x toppings and we only wanted groups of two different toppings, we would have x toppings for the first option, and x-1 toppings for the second option since we've already used a topping. Using the multiplication rule that would give us x(x-1) different groups, if order mattered, which is written as:

$_xP_2=\frac{x!}{(x-2)!}=x(x-1)$

So, in this specific problem order does not matter. We see that we've overcounted because for every pair of toppings we have, they can be reordered 2!=2 ways. So that means we must divide by two to get:

$\frac{x(x-1)}{2}$

Which is written as:

$_xC_2=\binom{x}{2}=\frac{x!}{(x-2)!2!}$