Question

A pizza place offers 3 different

cheeses and 8 different toppings. In how many ways can a pizza be made
with 1 cheese and 3 toppings?

Answer 1

`1\ cheese\ is\ choose\ from\ 3:\ \ \ {3 \choose 1} =3\\and\ 3\ toppings\ are\ choose\ from\ 8:\ \ \ {8 \choose 3} = (8!)/(3!\cdot(8-3)!)= (5!\cdot6\cdot7\cdot8)/(2\cdot3\cdot5!)=56 \\\\a\ pizza\ can\ be\ made\ on\ \ 3 \cdot 56=168\ \ ways`

Answer 2

The cheese can be any one out of 3 = 3 choices. For each of those . . .
The first topping can be any one of 8. For each of those . . .
The second topping can be any one of the remaining 7 . For each of those . . .
The third topping can be any one of the remaining 6 .

Total number of ways to assemble a pizza = (3 x 8 x 7 x 6) = 1,008 ways.

BUT . . .

You could choose the SAME 3 toppings in (3 x 2) = 6 different ways, and
nobody could tell the difference once they were selected. All 6 different
ways would result in the same pizza.

So, out of the 1,008 total different ways there are of choosing ingredients,
there are only ( 1,008 / 6 ) = 168 different and distinct pizzas.