# Mathematicians, Is this even possible?

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**2**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

Easily. There are 39 one-topping pizzas.

There are 39x39 = 1521 two-topping pizzas, including two of the same toppings.

39x39x39 = 59,319 three-topping pizzas.

39x39x39x39 = 2,313,441 four-topping pizzas.

Four toppings is halfway there, exponentially speaking. Eight-toppings is 5 trillion.

There are 39x39 = 1521 two-topping pizzas, including two of the same toppings.

39x39x39 = 59,319 three-topping pizzas.

39x39x39x39 = 2,313,441 four-topping pizzas.

Four toppings is halfway there, exponentially speaking. Eight-toppings is 5 trillion.

*Last edited by Nick Danger; 03-01-20 at 07:45 AM.*

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Bandoman (03-03-20)

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**Re: Mathematicians, Is this even possible?**

There are 39x39 = 1521 two-topping pizzas, including two of the same toppings.

39x39x39 = 59,319 three-topping pizzas.

39x39x39x39 = 2,313,441 four-topping pizzas.

Four toppings is halfway there, exponentially speaking. Eight-toppings is 5 trillion.

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**6**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

1,672,219,813,000 ish. (assuming no doubles, triples, etc. for each topping)

39 options

1 combo of 0,

39 combos of 1,

714 combos of 2,

etc.

39 options

1 combo of 0,

39 combos of 1,

714 combos of 2,

etc.

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**7**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

Double pepperoni is a two-topping pizza for the sake of the calculation. People do order it. Besides, I'm too lazy to do the permutation of 39 toppings taken 8 at a time. It's 39!/(39-8)! if I remember my high school math.

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**8**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

Once you get half way (19 toppings) the permutations start to go down again too, because of this.

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**9**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

So how many toppings does it take to reach one trillion? Do you have to go all the way to "one with everything"?

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**10**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

1

39

741

9139

82251

575757

3262623

15380937

61523748

211915132

635745396

1676056044

3910797436

8122425444

15084504396

25140840660

37711260990

51021117810

62359143990

68923264410

68923264410

62359143990

51021117810

37711260990

25140840660

15084504396

8122425444

3910797436

1676056044

635745396

211915132

61523748

15380937

3262623

575757

82251

9139

741

39

1

Note that the permutations in the middle may not be 100% accurate for the last digit. They end in 0 because of how they're displayed on my calculator (ie: 6.892326441 E+10)

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**11****Re: Mathematicians, Is this even possible?**

Don‘t forget that you can (at most pizza places) split the pizza in half and do different toppings on each half. So it’s every combination ever, and then you add that with every combination ever, permutated with every other combination ever.

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**13**DVD Talk Hero

**Re: Mathematicians, Is this even possible?**

If you had 39 toppings available, wouldn't the number of combinations be 2^39?

Think of a pizza like a binary number:

000000000000000000000000000000000000000

where a zero represents the absence of a topping and a one represents the topping on the pizza. If they don't limit the number of toppings per pizza and you can get as many as you want (which, with 39 options would be more like a big pile of meats and vegetables on a piece of bread), then it works out to a bit over 500 billion -- not counting double portions or half-pizzas.

But that numbers will launch into the stratosphere when start doing double and half-toppings.

Think of a pizza like a binary number:

000000000000000000000000000000000000000

where a zero represents the absence of a topping and a one represents the topping on the pizza. If they don't limit the number of toppings per pizza and you can get as many as you want (which, with 39 options would be more like a big pile of meats and vegetables on a piece of bread), then it works out to a bit over 500 billion -- not counting double portions or half-pizzas.

But that numbers will launch into the stratosphere when start doing double and half-toppings.

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**17**Moderator

**Re: Mathematicians, Is this even possible?**

Rough math and Mathematica says the correct answer is 549,755,813,887

Formula: 39!/1!(38!)+39!/2!(37!)....39!/39!(0!)

It is an N choose K formula, but you need to sum each specific instance of K.

Formula: 39!/1!(38!)+39!/2!(37!)....39!/39!(0!)

It is an N choose K formula, but you need to sum each specific instance of K.

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**18****Re: Mathematicians, Is this even possible?**

If p is the number of possible toppings, then the number of different possible combinations is 2^p. In order to have one trillion possible combinations, you would need to have at least 40 different possible toppings, because 40 is the smallest number p where (2^p) >= one trillion.

That list has 39 toppings.

So with 39 possible toppings, no, it is not possible to have one trillion different combinations.

If there were 40 toppings, then yes, it would be possible to have one trillion different combinations.

That list has 39 toppings.

So with 39 possible toppings, no, it is not possible to have one trillion different combinations.

If there were 40 toppings, then yes, it would be possible to have one trillion different combinations.

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**20**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

So if you allow for half-and-half pizzas, the total is 500 billion times 500 billion.

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**21**DVD Talk Hero

**Re: Mathematicians, Is this even possible?**

Yeah, we're looking at something like 300 sextillion different pizzas if we're half-and-halfing them.

In scientific notation, that's 3x10^23.

300,000,000,000,000,000,000,000.

In scientific notation, that's 3x10^23.

300,000,000,000,000,000,000,000.

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**22**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

Okay, I barely made it through ninth grade algebra and I am now sharpening a slide rule in preparation for cutting my own throat. Goodbye, cruel world.

0101001

0101001

**The following 2 users liked this post by Vibiana:**

Ash Ketchum (03-01-20),
Bandoman (03-03-20)

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**24**DVD Talk Legend

**Re: Mathematicians, Is this even possible?**

Looks like we have a few people here who have a better memory of Finite than I do. It was the most practical math course I ever took, but it was an elective, and after a few weeks I was the only one left in the class.