Number of ways to order a pizza You walk into a pizzeria and the deal for today is:
2 pizza
up to 5 toppings on each
11 toppings to choose from
all for $10

What is the total number of ways to order a pizza with up to 5 toppings when choosing from 11 toppings?
What I have so far:
The number of ways to order a pizza with 0 toppings: c(11, 0) = 1
The number of ways to order a pizza with 1 toppings: c(11, 1) = 11
The number of ways to order a pizza with 2 toppings: c(11, 2) = 55
The number of ways to order a pizza with 3 toppings: c(11, 3) = 165
The number of ways to order a pizza with 4 toppings: c(11, 4) = 330
The number of ways to order a pizza with 5 toppings: c(11, 5) = 462

Therefore, number of ways the pizza can be ordered is : 1024 ways.
So, by that logic, the second pizza can be ordered in 1024 ways as well:
so, 1024 x 1024 = $$1,048,576$$
Have I done this correctly?
 A: Yes, your reasoning and answer are correct. You could have avoided most of the arithmetic by noticing that since $\binom{11}k=\binom{11}{11-k}$,
$$\binom{11}0+\binom{11}1+\ldots+\binom{11}5=\binom{11}{11}+\binom{11}{10}+\ldots+\binom{11}6\;,\tag{1}$$
so that each side of $(1)$ must be exactly half of $$\sum_{k=0}^{11}\binom{11}k=2^{11}\;,$$ or $2^{10}=1024$. (I don’t say that you should have seen this shortcut, just that it does exist.)
A: The order of your pizza requests may not be relevant, so asking for a pepperoni and hawaiian pizza is the same as asking for a hawaiian and a papperoni. In that case the total number of options is $\frac{1024\cdot 1023}{2} + 1024 = \frac{1025\cdot 1024}{2}$
But, the question is rather explicit:

What is the total number of ways to order a pizza with up to 5 toppings when choosing from 11 toppings?

So, order one pizza. The fact that the commercial offer includes two pizza is irrelevent to this question.
A: You can have zero through five toppings on each pizza.  This gives
$$1 + _{11}C_1 + _{11}C_2 + _{11}C_3 + _{11}C_4 + _{11}C_5$$
ways to top one pizza.
Square this number for two pizzas.
So yes, your answer is correct.
