Pete's Pizzeria offers 30 different pizza toppings.
How many 3 topping pizzas are possible (assuming the toppings are distinct)?
If we allow any number of toppings (all distinct), including none, how many different pizzas are possible?
C(30,3) = 4060
any number means some subset of the power set of 30 elements. That is
2^30 = 1,073,741,824
To find the number of 3-topping pizzas, we can use the combination formula.
The combination formula is given by:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of items and r is the number of items chosen at a time.
In this case, we have 30 toppings to choose from and we want to choose 3 toppings for each pizza. So n = 30 and r = 3.
Plugging these values into the combination formula, we get:
C(30, 3) = 30! / (3!(30-3)!)
Calculating this expression, we find:
C(30, 3) = 30! / (3! * 27!)
Now, let's find the number of different pizzas possible when any number of toppings, including none, are allowed.
For each topping, there are two possibilities: either it is included on the pizza or it is not. Since there are 30 toppings in total, for each topping, we have 2 choices. Therefore, the total number of different pizzas possible is:
2^30
Hence, there are C(30, 3) = 4060 different 3-topping pizzas possible, and 2^30 = 1,073,741,824 different pizzas possible when any number of toppings, including none, are allowed.