In a pizza restaurant, you can get a basic pizza with two toppings, cheese and tomato. You can also make up your own pizza with extra toppings. You ca choose from four different extra toppings: olives, ham, mushrooms, and salami. Ross wants to order a pizza with different extra toppings. How many different combinations can Ross choose from?

You can use combinations to get to the total count of toppings you can make. Ross can make total of 15 different extra topping pizzas.

How many ways k things out of m different things (m ≥ k) can be chosen if order of the chosen things doesn't matter?

This can be done in total of [tex]^mC_k = \dfrac{m!}{k! \times (m-k)!}[/tex] ways.

For this case, since there are 4 types of extra toppings available and Ross can take any number of toppings.

If he chooses to get only 1 topping, then total [tex]^4C_1 = 4[/tex] ways of getting such pizzas.
If he chooses to get 2 different toppings, then total [tex]^4C_2 = \dfrac{4 \times 3}{2 \times 1} = 6[/tex] ways.
If he chooses to get 3 different toppings, then total [tex]^4C_3 = \dfrac{4 \times 3 \times 2}{3 \times 2 \times 1} = 4[/tex] ways.
If he chooses to get 4 different toppings, then total [tex]^4C_4 = \dfrac{4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} = 1[/tex] way.

Thus, in total there are 4+6+4+1 = 15 ways of getting pizzas with different extra toppings.

Thus, Ross can make total of 15 different extra topping pizzas.

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