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Wording questions regarding the following question:

"You are ordering two pizzas. A pizza can be small, medium, large or extra large, with any combination of 8 possible toppings (getting no toppings is also allowed, as is getting all of 8). How many possibilities are there for your two pizzas?"

The above question and similar questions have being asked here in the past. And the numerical answers are as such:

size choices = 4
topping choices = 9 (including 0 topping)

size choices = $\binom{4}{1}$
topping choices = $\sum_{i=0}^8\binom{8}{i}$

multiplication rule: $\binom{4}{1}$ $\sum_{i=0}^8\binom{8}{i} = 4*256 = 1024$

$\require{cancel}$ First question:
Is it necessary to $\cancel{multiple}$ square 1024 even when both pizzas has the same combinations to address the original question?

First question followup:
Following angryavian comment about squaring 1024, I went back to reread the answers and found this: "It seems more likely that getting pizza A and then B is the same as getting B and then A. In that case you have to divide the cases of disparate pizzas by 2, so there would be 1024 (ways to get two the same) + $\frac{1}{2}$⋅1024⋅1023 (ways to get two different)."

Does this implies to (1024 * $\frac{1}{2}$) * (1024-1); which is first to divide the over-count of 1024 then multiple 1024-1 which is to account for the 1 that is selected prior?

Second question:
I've seen an answer where $2^8$ is used instead of $\sum_{i=0}^8\binom{8}{i}$, where the contributor wrote that "the number of possible combinations of toppings is the same as sampling from the set {0,1} with replacement and with ordering, 8 times; there are 2^8=256 possible toppings combinations". Which I don't understand where he derived "8 times". And am I right to assume "set {0,1}" means 0 = with, 1 = without topping and vice versa.

Third question:
There are also mentioning of using the Einstein-Bose approach for replacement where $\binom{1024+2-1}{2}$. Since the question has no mentioning of replacement, hence, should I be viewing this as a with or without replacement question?

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    $\begingroup$ For your second question, your interpretation of $\{0, 1\}$ is correct: for each of the 8 toppings, you decide whether to include it or not (2 choices for each topping, $2^8$ for all 8 toppings). $\endgroup$
    – angryavian
    May 1, 2022 at 15:29
  • $\begingroup$ For your first question, you should be squaring 1024, not multiplying by 2. $\endgroup$
    – angryavian
    May 1, 2022 at 15:30
  • $\begingroup$ @angryavian, thank you for the reply, after seeing your comments, I've went back and reread some of the answers and found one that requires dividing 1024 by half and multiplying 1024-1. I've added it in the question. If possible could you also provide some insight on it? $\endgroup$ May 1, 2022 at 17:30
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    $\begingroup$ If the pizzas are distinguishable (e.g. one pizza for Alice, and one for Bob), then $1024^2$ is the answer. If you instead only care about the set of two pizzas (swapping them doesn't count as a new outcome), then there are $1024$ outcomes where both pizzas are the same, and $\binom{1024}{2} = \frac{1}{2}\cdot 1024 \cdot 1023$ outcomes where the two pizzas are different. $\endgroup$
    – angryavian
    May 1, 2022 at 17:46

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