Question

A pizza parlor offers 10 toppings. (a) How many 8-topping pizzas could they put on their menu, assuming double toppings are not allowed? (b) How many total pizzas are possible, with between zero and 10 toppings?

Answer 1

To put 8-topping pizzas on the menu, use combinations to select 8 toppings out of 10 available, resulting in 45 different pizzas. To find the total number of pizzas with between zero and 10 toppings, calculate the number of pizzas with each possible number of toppings and sum them up.

Step-by-step explanation:

To find the number of 8-topping pizzas that can be put on the menu without double toppings, we need to select 8 toppings out of the 10 available. This can be done using combinations. The formula for combinations is given by nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items selected.

Using this formula, we can calculate the number of 8-topping pizzas as follows:

1. Calculate 10C8: 10! / (8!(10-8)!)
2. Simplify the expression: 10! / (8! * 2!)
3. Further simplify: (10 * 9 * 8!) / (8! * 2)
4. Cancel out the 8! terms: (10 * 9) / 2
5. Calculate the final answer: 90 / 2 = 45

Therefore, the pizza parlor can put 45 different 8-topping pizzas on their menu without double toppings.

To find the total number of pizzas possible with between zero and 10 toppings, we can sum up the number of pizzas with each possible number of toppings:

1. Calculate 10C0: 10! / (0!(10-0)!)
2. Calculate 10C1: 10! / (1!(10-1)!)
3. Calculate 10C2: 10! / (2!(10-2)!)
4. Calculate 10C3: 10! / (3!(10-3)!)
5. Continue this calculation for 10C4, 10C5, 10C6, 10C7, 10C8, 10C9, and 10C10
6. Sum up all the results to find the total number of pizzas