EXAMPLE 1.5.14
A pizzeria is offering a special: for $6 you get a four-topping pizza. The choices for toppings are pepperoni, sausage, olives, mushrooms, anchovies, peppers, and onions. How many different 4-topping combinations are possible (assuming that no topping can be repeated on a pizza)?
SOLUTION
In this case, a combination of toppings is determined entirely by which four items are chosen, and not by the order in which they are chosen or listed (for instance, if you order a pizza with pepperoni, sausage, olives, and mushrooms it's the same as ordering a pizza with mushrooms, olives, sausage, and pepperoni; rearranging the order in which the four elements are listed does not give a different outcome). The number of 4-topping combinations is the number of combinations of 7 things taken 4 at a time.
There are 35 ways to combine four of the toppings.