Hello everyone, I'm just someone you may have forgotten about. You may recall I was recommended to be a Moderator here by AskJoe and for my instalment of Forum News.

In Papa's Taco Mia! (Desktop), there are 18,103,164 possible tacos that can be ordered*.

*Ordered is defined as being logically written on an order ticket. For example, you can have five consecutive sour cream servings (although that would be extremely disgusting), but you can't have sour cream - blank space - pinto beans.

If each taco took exactly two minutes to make, and you made only one taco at a time, it would take 68 years*, 323 days, 6 hours, and 48 minutes of non-stop taco making to create all of the possible combinations.

*assuming a year is always exactly 365 days

I'm going to try and calculate this for Scooperia, but it's much more complicated for several reasons:

- Each order can have either 1, 2, or 3 scoops, each with their own mixable and dough as well as ice cream. - Each order can have up to 3 placeable toppings, depending on the amount of scoops. - If an order has a banana, it can have one pourable topping less. - If an order has no placeable toppings, it can have one pourable topping more. - An order can have seasonal ingredients, but only from one holiday at a time. For example, you can't have a candy rocket and caramel apple ice cream on the same order. - Holiday ingredients usually consist of 1 ice cream, 2 mixables, 1 placeable and 2 pourables, but not always. - Big Top Carnival's placeable is a type of banana, which behaves differently than other placeables.

Well, wish me luck!

« Last Edit: January 12, 2020, 03:50:37 PM by Ninja Monkey »

Logged

"Most people don't know why I do the things I do. Most people don't even know my real name. I'd like to keep it that way." -E.N.I.G.M.A.

I'm going to try and calculate this for Scooperia, but it's much more complicated for several reasons:

- Each order can have either 1, 2, or 3 scoops, each with their own mixable and dough as well as ice cream. - Each order can have up to 3 placeable toppings, depending on the amount of scoops. - If an order has a banana, it can have one pourable topping less. - If an order has no placeable toppings, it can have one pourable topping more. - An order can have seasonal ingredients, but only from one holiday at a time. For example, you can't have a candy rocket and caramel apple ice cream on the same order. - Holiday ingredients usually consist of 1 ice cream, 2 mixables, 1 placeable and 2 pourables, but not always. - Big Top Carnival's placeable is a type of banana, which behaves differently than other placeables.

For now, I'll focus on the sundaes without holiday ingredients first.

I'll start with calculating the amount of possibilities per scoop, so I'll discount any pourables or bananas.

Each scoop has a dough, a mixable, an ice cream and possibly a placeable. Since having no placeables affects the rest of the sundae, I'll calculate the placeables later.

Per scoop = amount of doughs * amount of mixables * amount of ice creams = 8*19*19=2888

This means that a two-scooper would have 2888*2888=8,340,544 possibilities, while a three-scooper would have 2888*2888*2888=24,087,491,072 possibilities! And that's without even counting any toppings, or seasonal ingredients!

For now, let's assume that the sundaes have no placeables or bananas, since that makes things a lot simpler. Since we don't have per-scoop pourables, it doesn't matter how many scoops we have.

Without placeables or bananas, we can have up to 5 pourables. Amount of pourable possibilities = (p^{5}+p^{4}+p^{3}+p^{2}+p+1), where p is the total amount of pourable toppings available. In this case, it's 12^{5}+12^{4}+12^{3}+12^{2}+12+1=271453 possibilities.

So, for 1-scoop sundaes without placeables or bananas, we have 2888*271453=783,956,264 possibilities. For 2-scoopers that's 8340544*271453=2,264,065,690,432, and for 3-scoopers a whopping 24087491072*271453=6,538,621,713,967,616 possibilities.

Now say we have a banana, but no placeables. This means we have at most 4 pourables, leaving us with 12^{4}+12^{3}+12^{2}+12+1=22621 possibilities for pourables. Doing the math, that's 2888*22621=65,329,448 for 1-scoopers, 8340544*22621=188,671,445,824 for 2-scoopers, and 24087491072*22621=544,883,135,539,712 for 3-scoopers.

If you want placeables instead of a banana, it's largely the same except you now need to deal with the amount of placeable toppings too. All in all, there are 8 standard placeable toppings, not counting bananas since they work differently. This means there are 9 possibilities per scoop, because it's also possible for that particular scoop to have no placeables. For 1-scoop sundaes, you've simply got 8 possibilities. For 2-scoop sundaes, you have 9*9-1=80 possibilities. For 3-scoops, 9*9*9-1=728 possibilities. Note that I removed the ones where all 3 come up empty.

This means that, if you have at least 1 placeable, there's 65329448*8=522,635,584 1-scoopers, 188671445824*80=15,093,715,665,920 2-scoopers and 544883135539712*728=396,674,922,672,910,336 3-scoopers.

There's one more possibility: having both placeables and a banana. This is largely the same, except the amount of pourables is only 12^{3}+12^{2}+12+1=1885. This means that 1-scoopers are 2888*1885*8=43551040, 2-scoopers are 8340544*1885*80=1257754035200 and 3-scoopers are 24087491072*1885*728=33054782248284160.

This means that the total amount of purely standard-ingredient sundaes is 783956264+2264065690432+6538621713967616+65329448+188671445824+544883135539712+522635584+15093715665920+396674922672910336+43551040+1257754035200+33054782248284160=436832015393011536, or about 437 quadrillion. And we haven't even gotten to the holiday ingredients yet!

For most holidays, you have 1 extra icecream, 2 extra mixables, 1 extra placeable and 2 extra pourables.

This means that, during a typical holiday, the per-scoop count is 8*21*20=3360. That means two scoops are 3360*3360=11289600, and 3 are 3360*3360*3360=37933056000.

The amount of possibilities for up to 5 pourables is up as well: 14^{5}+14^{4}+14^{3}+14^{2}+14+1=579195

For 1 scoop without placeables, that's 3360*579195=1946095200. For 2 it's 11289600*579195=6538879872000, and for 3 it's 37933056000*579195=21970636369920000

For sundaes with a banana, the pourables are 14^{4}+14^{3}+14^{2}+14+1=41371, and the total amount of possibilities 3360*41371=139006560, 11289600*41371=467062041600, and 37933056000*41371=1569328459776000 respectively.

With a placeable, it's that times 9 for single-scoop, times 10*10-1=99 for double scoop, and times 10*10*10-1=999 for triple scoop, coming to a total of 139006560*9=1251059040, 467062041600*99=46239142118400 and 1569328459776000*999=1567759131316224000 respectively.

With both a banana and placeables, 14^{3}+14^{2}+14+1=2955 and 3360*2955*9=89359200, 11289600*2955*99=3302716032000 and 37933056000*2955*999=111980088299520000 respectively.

This brings the total amount of holiday sundaes up to 1946095200+6538879872000+21970636369920000+139006560+467062041600+1569328459776000+1251059040+46239142118400+1567759131316224000+89359200+3302716032000+111980088299520000=1703335735671024000, or 1.7 quintillion.

To get the amount of sundaes with at least 1 holiday ingredient, we simply subtract the number of ‘standard’ sundaes, so 1703335735671024000-436832015393011536=1266503720278012464, or about 1.3 quintillion per holiday.

For Big Top Carnival, we need some special calculations. Instead of an extra placeable, we have an extra banana. If we don’t have placeables or a banana, this makes no difference whatsoever and the totals are still 1946095200, 6538879872000 and 21970636369920000.

For the ones with a banana and no placeables, we need to double the numbers, since each order can occur with either type of banana. This means the totals here are 139006560*2=278013120, 467062041600*2=934124083200 and 1569328459776000*2=3138656919552000.

For the orders with just placeables, we need to perform the same calculations, but with 8, 80 and 728 instead of 9, 99 and 999, coming to a total of 139006560*8=1112052480, 467062041600*80=37364963328000 and 1569328459776000*728=1142471118716928000.

For the ones with both a banana and placeables, we again need to perform the calculations with 8, 80 and 728, but we also need to double every number. This means we’re left with 3360*2955*8*2=158860800, 11289600*2955*80*2=5337722880000 and 37933056000*2955*728*2=163206214778880000.

The total amount of Big Top Carnival sundaes is 1946095200+6538879872000+21970636369920000+278013120+934124083200+3138656919552000+1112052480+37364963328000+1142471118716928000+158860800+5337722880000+163206214778880000=1330836805970464800, or 1.3 quintillion. Subtracting the standard sundaes gives 1330836805970464800-436832015393011536=894004790577453264, or 894 quadrillion.

This means that the final total is 436832015393011536+1266503720278012464*11+894004790577453264=15,262,377,729,028,601,904, or about 15 quintillion. To give you an idea of how ridiculously large that number is, if each order took exactly 2 minutes to prepare, and you only worked on 1 at a time, it would take about 58 trillion years to finish making them all, which is about 4200 times the age of the universe. Obviously, nobody has that much time, but apparently I had enough of that to calculate this.

Logged

"Most people don't know why I do the things I do. Most people don't even know my real name. I'd like to keep it that way." -E.N.I.G.M.A.

I'll take a Double Triple Bossy Deluxe on a raft, 4x4 animal style, extra shingles with a shimmy and a squeeze, light axle grease; make it cry, burn it, and let it swim.

For now, I'll focus on the sundaes without holiday ingredients first.

I'll start with calculating the amount of possibilities per scoop, so I'll discount any pourables or bananas.

Each scoop has a dough, a mixable, an ice cream and possibly a placeable. Since having no placeables affects the rest of the sundae, I'll calculate the placeables later.

Per scoop = amount of doughs * amount of mixables * amount of ice creams = 8*19*19=2888

This means that a two-scooper would have 2888*2888=8,340,544 possibilities, while a three-scooper would have 2888*2888*2888=24,087,491,072 possibilities! And that's without even counting any toppings, or seasonal ingredients!

For now, let's assume that the sundaes have no placeables or bananas, since that makes things a lot simpler. Since we don't have per-scoop pourables, it doesn't matter how many scoops we have.

Without placeables or bananas, we can have up to 5 pourables. Amount of pourable possibilities = (p^{5}+p^{4}+p^{3}+p^{2}+p+1), where p is the total amount of pourable toppings available. In this case, it's 12^{5}+12^{4}+12^{3}+12^{2}+12+1=271453 possibilities.

So, for 1-scoop sundaes without placeables or bananas, we have 2888*271453=783,956,264 possibilities. For 2-scoopers that's 8340544*271453=2,264,065,690,432, and for 3-scoopers a whopping 24087491072*271453=6,538,621,713,967,616 possibilities.

Now say we have a banana, but no placeables. This means we have at most 4 pourables, leaving us with 12^{4}+12^{3}+12^{2}+12+1=22621 possibilities for pourables. Doing the math, that's 2888*22621=65,329,448 for 1-scoopers, 8340544*22621=188,671,445,824 for 2-scoopers, and 24087491072*22621=544,883,135,539,712 for 3-scoopers.

If you want placeables instead of a banana, it's largely the same except you now need to deal with the amount of placeable toppings too. All in all, there are 8 standard placeable toppings, not counting bananas since they work differently. This means there are 9 possibilities per scoop, because it's also possible for that particular scoop to have no placeables. For 1-scoop sundaes, you've simply got 8 possibilities. For 2-scoop sundaes, you have 9*9-1=80 possibilities. For 3-scoops, 9*9*9-1=728 possibilities. Note that I removed the ones where all 3 come up empty.

This means that, if you have at least 1 placeable, there's 65329448*8=522,635,584 1-scoopers, 188671445824*80=15,093,715,665,920 2-scoopers and 544883135539712*728=396,674,922,672,910,336 3-scoopers.

There's one more possibility: having both placeables and a banana. This is largely the same, except the amount of pourables is only 12^{3}+12^{2}+12+1=1885. This means that 1-scoopers are 2888*1885*8=43551040, 2-scoopers are 8340544*1885*80=1257754035200 and 3-scoopers are 24087491072*1885*728=33054782248284160.

This means that the total amount of purely standard-ingredient sundaes is 783956264+2264065690432+6538621713967616+65329448+188671445824+544883135539712+522635584+15093715665920+396674922672910336+43551040+1257754035200+33054782248284160=436832015393011536, or about 437 quadrillion. And we haven't even gotten to the holiday ingredients yet!

For most holidays, you have 1 extra icecream, 2 extra mixables, 1 extra placeable and 2 extra pourables.

This means that, during a typical holiday, the per-scoop count is 8*21*20=3360. That means two scoops are 3360*3360=11289600, and 3 are 3360*3360*3360=37933056000.

The amount of possibilities for up to 5 pourables is up as well: 14^{5}+14^{4}+14^{3}+14^{2}+14+1=579195

For 1 scoop without placeables, that's 3360*579195=1946095200. For 2 it's 11289600*579195=6538879872000, and for 3 it's 37933056000*579195=21970636369920000

For sundaes with a banana, the pourables are 14^{4}+14^{3}+14^{2}+14+1=41371, and the total amount of possibilities 3360*41371=139006560, 11289600*41371=467062041600, and 37933056000*41371=1569328459776000 respectively.

With a placeable, it's that times 9 for single-scoop, times 10*10-1=99 for double scoop, and times 10*10*10-1=999 for triple scoop, coming to a total of 139006560*9=1251059040, 467062041600*99=46239142118400 and 1569328459776000*999=1567759131316224000 respectively.

With both a banana and placeables, 14^{3}+14^{2}+14+1=2955 and 3360*2955*9=89359200, 11289600*2955*99=3302716032000 and 37933056000*2955*999=111980088299520000 respectively.

This brings the total amount of holiday sundaes up to 1946095200+6538879872000+21970636369920000+139006560+467062041600+1569328459776000+1251059040+46239142118400+1567759131316224000+89359200+3302716032000+111980088299520000=1703335735671024000, or 1.7 quintillion.

To get the amount of sundaes with at least 1 holiday ingredient, we simply subtract the number of ‘standard’ sundaes, so 1703335735671024000-436832015393011536=1266503720278012464, or about 1.3 quintillion per holiday.

For Big Top Carnival, we need some special calculations. Instead of an extra placeable, we have an extra banana. If we don’t have placeables or a banana, this makes no difference whatsoever and the totals are still 1946095200, 6538879872000 and 21970636369920000.

For the ones with a banana and no placeables, we need to double the numbers, since each order can occur with either type of banana. This means the totals here are 139006560*2=278013120, 467062041600*2=934124083200 and 1569328459776000*2=3138656919552000.

For the orders with just placeables, we need to perform the same calculations, but with 8, 80 and 728 instead of 9, 99 and 999, coming to a total of 139006560*8=1112052480, 467062041600*80=37364963328000 and 1569328459776000*728=1142471118716928000.

For the ones with both a banana and placeables, we again need to perform the calculations with 8, 80 and 728, but we also need to double every number. This means we’re left with 3360*2955*8*2=158860800, 11289600*2955*80*2=5337722880000 and 37933056000*2955*728*2=163206214778880000.

The total amount of Big Top Carnival sundaes is 1946095200+6538879872000+21970636369920000+278013120+934124083200+3138656919552000+1112052480+37364963328000+1142471118716928000+158860800+5337722880000+163206214778880000=1330836805970464800, or 1.3 quintillion. Subtracting the standard sundaes gives 1330836805970464800-436832015393011536=894004790577453264, or 894 quadrillion.

This means that the final total is 436832015393011536+1266503720278012464*11+894004790577453264=15,262,377,729,028,601,904, or about 15 quintillion. To give you an idea of how ridiculously large that number is, if each order took exactly 2 minutes to prepare, and you only worked on 1 at a time, it would take about 58 trillion years to finish making them all, which is about 4200 times the age of the universe. Obviously, nobody has that much time, but apparently I had enough of that to calculate this.

I'll take a Double Triple Bossy Deluxe on a raft, 4x4 animal style, extra shingles with a shimmy and a squeeze, light axle grease; make it cry, burn it, and let it swim.

For now, I'll focus on the sundaes without holiday ingredients first.

I'll start with calculating the amount of possibilities per scoop, so I'll discount any pourables or bananas.

Each scoop has a dough, a mixable, an ice cream and possibly a placeable. Since having no placeables affects the rest of the sundae, I'll calculate the placeables later.

Per scoop = amount of doughs * amount of mixables * amount of ice creams = 8*19*19=2888

This means that a two-scooper would have 2888*2888=8,340,544 possibilities, while a three-scooper would have 2888*2888*2888=24,087,491,072 possibilities! And that's without even counting any toppings, or seasonal ingredients!

For now, let's assume that the sundaes have no placeables or bananas, since that makes things a lot simpler. Since we don't have per-scoop pourables, it doesn't matter how many scoops we have.

Without placeables or bananas, we can have up to 5 pourables. Amount of pourable possibilities = (p^{5}+p^{4}+p^{3}+p^{2}+p+1), where p is the total amount of pourable toppings available. In this case, it's 12^{5}+12^{4}+12^{3}+12^{2}+12+1=271453 possibilities.

So, for 1-scoop sundaes without placeables or bananas, we have 2888*271453=783,956,264 possibilities. For 2-scoopers that's 8340544*271453=2,264,065,690,432, and for 3-scoopers a whopping 24087491072*271453=6,538,621,713,967,616 possibilities.

Now say we have a banana, but no placeables. This means we have at most 4 pourables, leaving us with 12^{4}+12^{3}+12^{2}+12+1=22621 possibilities for pourables. Doing the math, that's 2888*22621=65,329,448 for 1-scoopers, 8340544*22621=188,671,445,824 for 2-scoopers, and 24087491072*22621=544,883,135,539,712 for 3-scoopers.

If you want placeables instead of a banana, it's largely the same except you now need to deal with the amount of placeable toppings too. All in all, there are 8 standard placeable toppings, not counting bananas since they work differently. This means there are 9 possibilities per scoop, because it's also possible for that particular scoop to have no placeables. For 1-scoop sundaes, you've simply got 8 possibilities. For 2-scoop sundaes, you have 9*9-1=80 possibilities. For 3-scoops, 9*9*9-1=728 possibilities. Note that I removed the ones where all 3 come up empty.

This means that, if you have at least 1 placeable, there's 65329448*8=522,635,584 1-scoopers, 188671445824*80=15,093,715,665,920 2-scoopers and 544883135539712*728=396,674,922,672,910,336 3-scoopers.

There's one more possibility: having both placeables and a banana. This is largely the same, except the amount of pourables is only 12^{3}+12^{2}+12+1=1885. This means that 1-scoopers are 2888*1885*8=43551040, 2-scoopers are 8340544*1885*80=1257754035200 and 3-scoopers are 24087491072*1885*728=33054782248284160.

This means that the total amount of purely standard-ingredient sundaes is 783956264+2264065690432+6538621713967616+65329448+188671445824+544883135539712+522635584+15093715665920+396674922672910336+43551040+1257754035200+33054782248284160=436832015393011536, or about 437 quadrillion. And we haven't even gotten to the holiday ingredients yet!

For most holidays, you have 1 extra icecream, 2 extra mixables, 1 extra placeable and 2 extra pourables.

This means that, during a typical holiday, the per-scoop count is 8*21*20=3360. That means two scoops are 3360*3360=11289600, and 3 are 3360*3360*3360=37933056000.

The amount of possibilities for up to 5 pourables is up as well: 14^{5}+14^{4}+14^{3}+14^{2}+14+1=579195

For 1 scoop without placeables, that's 3360*579195=1946095200. For 2 it's 11289600*579195=6538879872000, and for 3 it's 37933056000*579195=21970636369920000

For sundaes with a banana, the pourables are 14^{4}+14^{3}+14^{2}+14+1=41371, and the total amount of possibilities 3360*41371=139006560, 11289600*41371=467062041600, and 37933056000*41371=1569328459776000 respectively.

With a placeable, it's that times 9 for single-scoop, times 10*10-1=99 for double scoop, and times 10*10*10-1=999 for triple scoop, coming to a total of 139006560*9=1251059040, 467062041600*99=46239142118400 and 1569328459776000*999=1567759131316224000 respectively.

With both a banana and placeables, 14^{3}+14^{2}+14+1=2955 and 3360*2955*9=89359200, 11289600*2955*99=3302716032000 and 37933056000*2955*999=111980088299520000 respectively.

This brings the total amount of holiday sundaes up to 1946095200+6538879872000+21970636369920000+139006560+467062041600+1569328459776000+1251059040+46239142118400+1567759131316224000+89359200+3302716032000+111980088299520000=1703335735671024000, or 1.7 quintillion.

To get the amount of sundaes with at least 1 holiday ingredient, we simply subtract the number of ‘standard’ sundaes, so 1703335735671024000-436832015393011536=1266503720278012464, or about 1.3 quintillion per holiday.

For Big Top Carnival, we need some special calculations. Instead of an extra placeable, we have an extra banana. If we don’t have placeables or a banana, this makes no difference whatsoever and the totals are still 1946095200, 6538879872000 and 21970636369920000.

For the ones with a banana and no placeables, we need to double the numbers, since each order can occur with either type of banana. This means the totals here are 139006560*2=278013120, 467062041600*2=934124083200 and 1569328459776000*2=3138656919552000.

For the orders with just placeables, we need to perform the same calculations, but with 8, 80 and 728 instead of 9, 99 and 999, coming to a total of 139006560*8=1112052480, 467062041600*80=37364963328000 and 1569328459776000*728=1142471118716928000.

For the ones with both a banana and placeables, we again need to perform the calculations with 8, 80 and 728, but we also need to double every number. This means we’re left with 3360*2955*8*2=158860800, 11289600*2955*80*2=5337722880000 and 37933056000*2955*728*2=163206214778880000.

The total amount of Big Top Carnival sundaes is 1946095200+6538879872000+21970636369920000+278013120+934124083200+3138656919552000+1112052480+37364963328000+1142471118716928000+158860800+5337722880000+163206214778880000=1330836805970464800, or 1.3 quintillion. Subtracting the standard sundaes gives 1330836805970464800-436832015393011536=894004790577453264, or 894 quadrillion.

This means that the final total is 436832015393011536+1266503720278012464*11+894004790577453264=15,262,377,729,028,601,904, or about 15 quintillion. To give you an idea of how ridiculously large that number is, if each order took exactly 2 minutes to prepare, and you only worked on 1 at a time, it would take about 58 trillion years to finish making them all, which is about 4200 times the age of the universe. Obviously, nobody has that much time, but apparently I had enough of that to calculate this.

I don't hate him. It's just that if I had to kill 15 Flipline customers, he would be the 6th one.

-PastelPenguins- [Apr 10 04:56 PM]: I mean, religion should have it's importance to everyone who has one but that shouldn't interfere with their view on other people

For now, I'll focus on the sundaes without holiday ingredients first.

I'll start with calculating the amount of possibilities per scoop, so I'll discount any pourables or bananas.

Each scoop has a dough, a mixable, an ice cream and possibly a placeable. Since having no placeables affects the rest of the sundae, I'll calculate the placeables later.

Per scoop = amount of doughs * amount of mixables * amount of ice creams = 8*19*19=2888

This means that a two-scooper would have 2888*2888=8,340,544 possibilities, while a three-scooper would have 2888*2888*2888=24,087,491,072 possibilities! And that's without even counting any toppings, or seasonal ingredients!

For now, let's assume that the sundaes have no placeables or bananas, since that makes things a lot simpler. Since we don't have per-scoop pourables, it doesn't matter how many scoops we have.

Without placeables or bananas, we can have up to 5 pourables. Amount of pourable possibilities = (p^{5}+p^{4}+p^{3}+p^{2}+p+1), where p is the total amount of pourable toppings available. In this case, it's 12^{5}+12^{4}+12^{3}+12^{2}+12+1=271453 possibilities.

So, for 1-scoop sundaes without placeables or bananas, we have 2888*271453=783,956,264 possibilities. For 2-scoopers that's 8340544*271453=2,264,065,690,432, and for 3-scoopers a whopping 24087491072*271453=6,538,621,713,967,616 possibilities.

Now say we have a banana, but no placeables. This means we have at most 4 pourables, leaving us with 12^{4}+12^{3}+12^{2}+12+1=22621 possibilities for pourables. Doing the math, that's 2888*22621=65,329,448 for 1-scoopers, 8340544*22621=188,671,445,824 for 2-scoopers, and 24087491072*22621=544,883,135,539,712 for 3-scoopers.

If you want placeables instead of a banana, it's largely the same except you now need to deal with the amount of placeable toppings too. All in all, there are 8 standard placeable toppings, not counting bananas since they work differently. This means there are 9 possibilities per scoop, because it's also possible for that particular scoop to have no placeables. For 1-scoop sundaes, you've simply got 8 possibilities. For 2-scoop sundaes, you have 9*9-1=80 possibilities. For 3-scoops, 9*9*9-1=728 possibilities. Note that I removed the ones where all 3 come up empty.

This means that, if you have at least 1 placeable, there's 65329448*8=522,635,584 1-scoopers, 188671445824*80=15,093,715,665,920 2-scoopers and 544883135539712*728=396,674,922,672,910,336 3-scoopers.

There's one more possibility: having both placeables and a banana. This is largely the same, except the amount of pourables is only 12^{3}+12^{2}+12+1=1885. This means that 1-scoopers are 2888*1885*8=43551040, 2-scoopers are 8340544*1885*80=1257754035200 and 3-scoopers are 24087491072*1885*728=33054782248284160.

This means that the total amount of purely standard-ingredient sundaes is 783956264+2264065690432+6538621713967616+65329448+188671445824+544883135539712+522635584+15093715665920+396674922672910336+43551040+1257754035200+33054782248284160=436832015393011536, or about 437 quadrillion. And we haven't even gotten to the holiday ingredients yet!

For most holidays, you have 1 extra icecream, 2 extra mixables, 1 extra placeable and 2 extra pourables.

This means that, during a typical holiday, the per-scoop count is 8*21*20=3360. That means two scoops are 3360*3360=11289600, and 3 are 3360*3360*3360=37933056000.

The amount of possibilities for up to 5 pourables is up as well: 14^{5}+14^{4}+14^{3}+14^{2}+14+1=579195

For 1 scoop without placeables, that's 3360*579195=1946095200. For 2 it's 11289600*579195=6538879872000, and for 3 it's 37933056000*579195=21970636369920000

For sundaes with a banana, the pourables are 14^{4}+14^{3}+14^{2}+14+1=41371, and the total amount of possibilities 3360*41371=139006560, 11289600*41371=467062041600, and 37933056000*41371=1569328459776000 respectively.

With a placeable, it's that times 9 for single-scoop, times 10*10-1=99 for double scoop, and times 10*10*10-1=999 for triple scoop, coming to a total of 139006560*9=1251059040, 467062041600*99=46239142118400 and 1569328459776000*999=1567759131316224000 respectively.

With both a banana and placeables, 14^{3}+14^{2}+14+1=2955 and 3360*2955*9=89359200, 11289600*2955*99=3302716032000 and 37933056000*2955*999=111980088299520000 respectively.

This brings the total amount of holiday sundaes up to 1946095200+6538879872000+21970636369920000+139006560+467062041600+1569328459776000+1251059040+46239142118400+1567759131316224000+89359200+3302716032000+111980088299520000=1703335735671024000, or 1.7 quintillion.

To get the amount of sundaes with at least 1 holiday ingredient, we simply subtract the number of ‘standard’ sundaes, so 1703335735671024000-436832015393011536=1266503720278012464, or about 1.3 quintillion per holiday.

For Big Top Carnival, we need some special calculations. Instead of an extra placeable, we have an extra banana. If we don’t have placeables or a banana, this makes no difference whatsoever and the totals are still 1946095200, 6538879872000 and 21970636369920000.

For the ones with a banana and no placeables, we need to double the numbers, since each order can occur with either type of banana. This means the totals here are 139006560*2=278013120, 467062041600*2=934124083200 and 1569328459776000*2=3138656919552000.

For the orders with just placeables, we need to perform the same calculations, but with 8, 80 and 728 instead of 9, 99 and 999, coming to a total of 139006560*8=1112052480, 467062041600*80=37364963328000 and 1569328459776000*728=1142471118716928000.

For the ones with both a banana and placeables, we again need to perform the calculations with 8, 80 and 728, but we also need to double every number. This means we’re left with 3360*2955*8*2=158860800, 11289600*2955*80*2=5337722880000 and 37933056000*2955*728*2=163206214778880000.

The total amount of Big Top Carnival sundaes is 1946095200+6538879872000+21970636369920000+278013120+934124083200+3138656919552000+1112052480+37364963328000+1142471118716928000+158860800+5337722880000+163206214778880000=1330836805970464800, or 1.3 quintillion. Subtracting the standard sundaes gives 1330836805970464800-436832015393011536=894004790577453264, or 894 quadrillion.

This means that the final total is 436832015393011536+1266503720278012464*11+894004790577453264=15,262,377,729,028,601,904, or about 15 quintillion. To give you an idea of how ridiculously large that number is, if each order took exactly 2 minutes to prepare, and you only worked on 1 at a time, it would take about 58 trillion years to finish making them all, which is about 4200 times the age of the universe. Obviously, nobody has that much time, but apparently I had enough of that to calculate this.