Steem Riddle of the Week #11: Two Decks -- Solution Revealed + Prize awared

For those who missed it, the full riddle is stated here: https://steemit.com/riddle/@droopy/steem-riddle-of-the-week-11-two-decks

Don't scroll down unless you don't mind spoilers!

Solution

The answer to the problem is quite close to 1-1/e or 0.632.

To solve this you need to calculate the number of "Derangements" which is the subset of the number of permutations of N items where none are in the original order. In combinatorics, often "derangements" of n items is denoted as !n.

Without loss of generality you can assume that one of the decks is in order a,b,c,d,e...

You then know that there are 52! possible orderings for the other deck, of which it turns out, about 1/e are derangements (no element is in its starting position) when n is reasonably large. To learn more about derangements, see here: https://en.wikipedia.org/wiki/Derangement

Prize

This was the first week in several months were no one won the prize!

Don't forget to follow and to check out my past riddles:

https://steemit.com/life/@droopy/steem-riddle-of-the-week-2-flipping-cards-prize-doubled
https://steemit.com/life/@droopy/steem-riddle-of-the-week-3-alternating-truth-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-4-black-cards-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-5-the-cup-game-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-6-burning-ropes-prize-doubled
https://steemit.com/life/@droopy/steem-riddle-of-the-week-7-midpoints
https://steemit.com/life/@droopy/steem-riddle-of-the-week-8-truth-lies-and-chaos
https://steemit.com/riddle/@droopy/steem-riddle-of-the-week-9-wandering-monster
https://steemit.com/riddle/@droopy/steem-riddle-of-the-week-10-weighted-balls