Erica started her own pizza restaurant. She offers one type of crust, one kind of sauce and 6 different toppings: sausage, pepperoni, mushroom, onions, green peppers and tomatoes.
1. How many different two-topping pizzas can she make?
2. Erica decides to offer three kinds of sauce: regular, spicy and extra hot. How many kinds of two-topping pizzas can she make?
3. Then Erica decides to offer two kids of crust: regular and gluten free. How many kinds of two-topping pizzas can she make?
1. number of 2-topping pizzas
= 6x5 or 30
2. Assuming one sauce and 2 topping,
number of pizzas = 2x6x5 = 60
3. Assuming the previous as well,
now we have
2 x 2 x 6 x 5 = 120
To calculate the number of different two-topping pizzas Erica can make, we can use the combination formula.
In general, the formula for calculating the number of combinations of selecting r items from a set of n items is given by:
C(n, r) = n! / (r!(n-r)!)
Where "!" represents the factorial function.
1. First, let's calculate the number of different two-topping pizzas Erica can make when she offers one type of crust, one kind of sauce, and 6 different toppings.
In this case, we have 6 toppings to choose from, and we need to select any 2 of them.
C(6, 2) = 6! / (2!(6-2)!) = 6! / (2!4!) = (6 * 5 * 4!) / (2! * 4!) = 15
Therefore, Erica can make 15 different two-topping pizzas.
2. Now, let's calculate the number of different two-topping pizzas Erica can make when she offers three kinds of sauce.
Since there are 3 kinds of sauce, we need to consider each kind of sauce separately and sum up the combinations.
For regular sauce:
C(6, 2) = 15
For spicy sauce:
C(6, 2) = 15
For extra hot sauce:
C(6, 2) = 15
Adding these up, we get: 15 + 15 + 15 = 45
Therefore, Erica can make 45 different kinds of two-topping pizzas with three kinds of sauce.
3. Finally, let's calculate the number of different two-topping pizzas Erica can make when she offers two kinds of crust.
Since there are 2 kinds of crust, we again need to consider each kind of crust separately and sum up the combinations.
For regular crust:
C(6, 2) = 15
For gluten-free crust:
C(6, 2) = 15
Adding these up, we get: 15 + 15 = 30
Therefore, Erica can make 30 different kinds of two-topping pizzas with two kinds of crust.