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
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.
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