Question

A pizza place offers 5 different cheeses and 10 different toppings. In how many ways can a pizza be made with 3 cheese and 6 toppings?

Answer 1

2100

Explanation:

We're talking about combinations - that is, we don't care in what order the toppings are chosen (much like we don't care in what order we are dealt a poker hand). The general formula is:

`C_(n,k)=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

We can calculate the way to pick 3 cheeses from a pool of 5 and we can also calculate the number of ways to pick 6 toppings from a pool of 10. We can then multiply these two numbers to find a total number of ways to do them together.

`C_(5,3)xxC_(10,6)=(5!)/((3!)(5-3)!)xx(10!)/((6!)(10-6)!)=>`

`=>(5xx4xx3!)/(3!xx2)xx(10xx9xx8xx7xx6!)/(6!xx4xx3xx2)`

`=>10xx(10xx9xx8xx7)/24=2100`