- #1

alec_tronn

- 29

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## Homework Statement

You can order a pizza with up to four toppings (repetitions allowed) from a set of 12 toppings. The order of the toppings is unimportant. How many different pizzas can you order? (To clarify: this UP TO four toppings, so a completely empty pizza is fair game, as is a pizza with just 1 topping on it).

## The Attempt at a Solution

I decided that there was 1 pizza with nothing on it, and 12 with one topping on it. Going from there, there are 12[tex]^{2}[/tex]/2! ways to make a pizza with 2 toppings, 12[tex]^{3}[/tex]/3! ways to make a pizza with 3 toppings, and 12[tex]^{4}[/tex]/4! ways to make a pizza with 4 toppings. This is 1237.

A different attempt was made as follows:

How many pizzas are there with unique toppings?

12 choose 4 (all different toppings) + 12 choose 3 (3 different toppings and 1 empty spot) + 12 choose 2 (2 different toppings and 2 empty spots) + 12 choose 1 (1 different toppings and 3 empty spots) + 12 choose 0 (all empty)

How many pizzas are there with 2 identical toppings:

12 choose 3 (2 identical, 2 different) + 12 choose 2 (2 identical, 1 different, 1 blank) + 12 choose 1 (2 identical, 2 blank)

How many pizzas are there with 3 identical toppings:

12 choose 2 (3 identical and 1 different) + 12 choose 1 (3 identical and 1 blank)

How many pizzas are there with 4 identical toppings

12 choose 1 (all four identical)

How many pizzas are there with 2 duplicates:

12 choose 2 (2 identical and another 2 of a different type identical)

Add them all up, and you get 1248.

Both ways seem logically sound to me, but they're 11 pizzas off. Any ideas?