A pizza parlor offers 8 toppings How many total pizzas are possible, with between zero and eight toppings (but not double toppings) allowed? The pizza parlor will list the 8 toppings in two equal-sized columns on their menu: How many ways can they arrange the toppings in the left column?
The Correct Answer and Explanation is:
Answers
- Total different pizzas that can be ordered: 256
- Ways to list the toppings in the left-hand column of the menu: 1 680
Why 256 pizzas?
Each of the eight toppings is an independent yes/no choice:
| Topping | Choice |
|---|---|
| Pepperoni | on / off |
| Mushrooms | on / off |
| … | … |
| Topping 8 | on / off |
For every topping you have 2 possibilities (include it or don’t).
Because the choices are independent, the Rule of Product says you multiply: 2×2×⋯×2 (8 factors) = 28=256.2 \times 2 \times \cdots \times 2 \; (\text{8 factors}) \; = \; 2^{8}=256.
The count already includes the “plain” pizza (zero toppings) and the “works” (all eight). No doubles are allowed, so there are no other possibilities—256 is the complete set.
Why 1 680 menu arrangements for the left column?
The menu will display the eight distinct toppings in two equal-sized columns of four items apiece. To decide what appears in the left column (and in which order from top to bottom) you must:
- Pick which four toppings go there.
- Decide their vertical order.
Those two steps are usually combined into a single “permutation” count. Imagine you start with all eight toppings in a bag and fill the four left-column slots one by one:
- First slot: any of the 8 toppings can go here.
- Second slot: 7 remain.
- Third slot: 6 remain.
- Fourth slot: 5 remain.
Multiply the choices: 8×7×6×5 = 8P4 = 1 680.8 \times 7 \times 6 \times 5 \;=\; 8P4 \;=\; 1\,680.
(Here 8P48P4 is the permutation notation “8 items taken 4 at a time”.)
Notice that we do not worry about the right column; once the left column is set, the remaining four toppings automatically occupy the right side (their own order can be decided later, but the question asked only about the left column).
Putting the numbers in context
- 256 possible pizzas means the shop can print a compact menu yet still let customers create a quarter-thousand distinct pies—plenty of variety without overwhelming the kitchen.
- 1 680 left-column arrangements give the owner lots of freedom in how the toppings look on the printed menu, allowing for aesthetic choices (grouping meats together, alternating colors, etc.) while still fitting neatly into two balanced columns.
