I understand the math behind it, I just can’t wrap my head around the fact that something so common and limited like a deck of cards can have more ways to be arranged than something so massive like the earth with all its oceans and mountains has atoms.
In my mind it would make more sense that even a little pond has more atoms than there are deck arrangements.
Could it be due to the fact that atoms have a lot of empty space in them?
In: 318
It’s about the same as the number of atoms in the entire Milky Way.
If you go up to 58-60 cards you get the number of atoms in the observable universe.
The number of atoms on Earth is astronomically huge; around 10^(50). That is an awful lot of atoms.
The number of ways to shuffle a deck of 52 cards is 52! or about 10^(68).
There isn’t really a trick to this, or a way to explain it other than to say that factorials (which tell us how many ways we can order something) get really big really quickly.
Each time you add a card you’re taking the current number of ways to shuffle it and *multiplying* that by the number of cards you have now, because the new card can go in any place in any of the existing orders.
And that is a lot of options.
You need about 48 cards to get more than the number of atoms on the Earth.
No trick. No weird things about how atoms work or the space between them. Just a counter-intuitive result because we’re not good at thinking about numbers this big.
The statement that cards are common is not relevant. The fact that cards are limited is not relevant as well, as we’re not counting cards.
I will not explain the math, as you’re saying you understand it. It’s great for a 5 year old.
There is more atoms in the universe (or in the pond) than we can imagine.
Also 54 different things each on 54 positions can form more different sequences that we can imagine.
At the end of the day the lesson is only how important it is to be able to rely on mathematics when dealing with things tou cannot intuitively assess correctly.
It’s a great skill, required not only in cards and atoms, but also in basic home budgeting for example.
The earths mass is approximately 6×10^24 kg. The average atomic mass is 40u, or 6.64×10^-26 kg. So the rough number of atoms on the earth is 9×10^50 atoms.
That part is pretty straightforward.
How many combinations are there in a deck of cards? The answer is 52! (The exclamation point standing for factorial, not like wow 52!)
52! = 52×51×50×…×3×2×1 = 8×10^67
Why so many? Well think about it. How many ways are there to arrange the deck when you keep the order of every card the same except you exchange the top one for a different card? 52 right? Now do that again except now you can exchange the top two cards for any other 2. The number of combinations there is 52×51. Its one less because the original 52 combinations already contained one of the orders that would occur when moving the second card as well. Keep going like this and the pattern continues. Allowing for the top 3 cards to be exchanged while keeping the rest in order would lead to 52×51×50. And so on like that. So the total number of combinations would be 52!
Take a used deck. Shuffle it. Think about the wildly crazy odds that you just shuffled it perfectly into numbers and suits in order, as if it was a brand new deck. Now realize that everytime you shuffle a deck, it is likely the only time in history that a deck is in that specific order.
Ever hear about how if you double a penny every day you get $1 million dollars after a month? It’s kind of like that in that the numbers get large quickly in a way that’s not intuitive.
If you have only five cards how many possibilities are there?
5 * 4 * 3 * 2 * 1 = 120. Seems reasonable.
If you have 10 cards how many possibilities are there?
10* 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800!
You doubled the number of cards but the number of possibilities grew incredibly larger! This is the nature of “factorials”
How about 20 cards?
20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 2.432902×10^18 which is:
2,432,902,000,000,000,000
And that’s barely half of a deck of cards! Let’s add six cards to make it half:
26! (26 factorial which is how we notate the above math) = 4.0329146×10^26
403,291,460,000,000,000,000,000,000
The difference between half a deck and a full deck?
52! is 8.0658175×10^67
or:
80,658,175,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
Edit: Fixed some errors because math is hard.
You’ve got 52 ways to arrange a deck by only changing the first card in the pile.
But for each of those 52 ways, there’s 51 ways to arrange the second card. 52×51 = 2,652 ways to arrange the deck with just the first two card positions.
The third card gives 50 more options. 2,652×50 = 132,000.
Fourth card: 132,000 x 49 = almost 6.5 million.
Fifth card: 6.5 million x 48 = 312 million.
6th card: 312 million x 47 gets you close to 15 billion.
15 billion options only 6 cards in. Now keep going for all 52.
That’s how.
I think you’re getting caught up in the comparison of two different things.
The ‘arrangements’ boosts the cards massively, but the Earth is just, like, there, and unboosted.
–
Let’s try this to regain perspective:
Imagine getting an aeroplane, and then making a list of every country, and you’ll travel to each country on earth.
Well, there are 195. Thats more than 52. So there are more possible iteneraries we can plan than there are ways to shuffle a deck of cards.
(It is way way way more, like 300 more digits than a number with ‘only’ 67 digits. Each digit making it 10 times more, so exponentially larger. Like each country past 52 means the list of possible iteneraties dwarfs the amount of ways to shuffle the deck, and we do that dwarfing over 100 times, each time dwarfing it by a larger factor than before.)
–
Or, imagine all the ways you could arrange the atoms on earth in a line. That would be probably have something like 50 septillion digits in it. Truly absurdly large.
–
The large number comes from the mathematics we’re doing, in order to come up with how many things we could *imagine* being possible.
However, the ‘atoms in the earth’ is not a list of all the possible imaginable things, but instead some actual things.
There is one deck of cards, and it is in one (1) arrangement. It *could* have many arrangements, and that potential number of *different* situations is large. But the deck of cards remains tiny and small physically, despite its mathematical potential.
How much information does it take to to pick any specific atom on Earth? You just say “I choose atom #7089896719367543239300239183376473481657021774306” and that’s it.
How much information does it take to pick a specific ordering of a deck of cards? You need to say “♠3 ♠5 ♦Q ♠9 ♠4 ♥10 ♠A ♦9 ♣4 ♥Q ♦K ♦7 ♣J ♣2 ♥7 ♣8 ♦2 ♥A ♣6 ♦3 ♠8 ♣5 ♣9 ♣Q ♣3 ♥6 ♥J ♣7 ♥8 ♠7 ♥4 ♣10 ♠2 ♥2 ♥K ♦6 ♦8 ♠J ♠6 ♦10 ♠10 ♦4 ♦J ♥5 ♦5 ♠Q ♥3 ♣A ♦A ♠K ♥9 ♣K”, and that requires more information than the above.
Start small.
1 card = 1 combination = (a)
Add one card and how many positions it can possibly be in.
2 cards = 2 combinations = 1×2 = (ab, ba)
Continue and see how it gets out of hand very swiftly.
3 cards = 6 combinations = 1x2x3 = (abc, acb, cab, bac, bca, cba)
Each number of cards provides another level of depth to that factorial calculation.
4 cards = 4! = 1x2x3x4 = 24 combinations
5 cards = 5! = 1x2x3x4x5 = 120
6 cards = 6! = 720
n cards = n!
The issue is that it becomes very difficult to grasp the scale at which factorials grow by.
One house of cards (13 cards) itself is very large.
13 cards = 13! = 6’227’020’800
This is only a quarter of the deck. Adding two more cards takes this from billions to trillions.
15 cards = 15! = 1’307’674’368’000
By this point we might as well start notating by magnitude rather than decimals.
15! = 1,307×10^13
16! = 2,092×10^14
17! = 3,557×10^15
By this point the exponent portion of scientific notation itself starts to ramp up faster and faster, leading to numbers that quickly become eyewateringly gargantuan. Moving beyond 52 cards to ten decks of cards or 520! and you have a number with 1189 digits:
520! = 1,761×10^1188
The main problem we have as humans is the inability to grasp scale. We almost have to “scale scales” to even express larger quantities. 520! In decimal would need to be written out as:
1761040341 7821111561 4601171098 7957854291 1478279626 8251583564 1902026635 0135316290 4925074528 9405517992 1749200005 2911460700 8515543885 0638417965 7281543248 4672840715 8333741072 4876868829 3028640496 7263861328 0090511006 1351548170 0246220249 8385102898 0182317268 9568139770 9425453375 0710104868 8685272896 7058125963 6249860475 3026067474 8364588470 9505625753 3018986923 3270021404 2981663219 2746245464 8287426080 1256152782 4480789474 6803821196 9705419149 2765207220 7588214838 8770717113 5570628153 6477814465 2417458191 6029042277 6966857721 1157141090 7822821839 4079265980 2623387773 7406575881 3337082961 0683213807 2133782927 7256701491 6522921467 6654388845 2889966132 3641698867 9986260143 9766132564 5202793456 6904658180 7830713431 9818775325 2557500708 3897838744 2729671889 1113203589 0748607857 4041862606 9545051663 6666042139 3444027899 8174513866 3191272624 3989647203 4558073404 8347637510 1139676265 9176106316 8892577208 2865466253 2799672036 9649293722 5332094609 0105491544 7110179290 3402188204 6954442464 9441464187 9321328704 5077984515 9720373136 5688705702 0004247897 0118285935 3616646946 7375916584 4683475607 6856514969 6000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 000000000
Before long, factorials can become so large that in decimal notation there wouldn’t be enough universe to print out the number in Times New Roman 10pt. This is an inaccurate description, but illustrative.
Pick one card, you have 1 in 52 chance of getting the one you picked. Pick two cards, you have a 1 in 51 chance of getting the one you picked. And so on, until you go through the whole deck. Hence, there are 52*51*50*…*3*2*1 different possible permutations or 52!=~8.066*10^67. I don’t know if that’s more than the number of atoms in the universe, but it’s a pretty fucking big number as it is 67 digits long.
Latest Answers