The question is: what is the probability that a shuffled deck will have two position-adjacent cards that are of the same suit and numerically adjacent (with A being considered adjacent to both 2 and K)? Any pair of such cards anywhere in the deck satisfies the requirement.
I thought I understood how to tackle this problem, which a buddy came up with (for fun), but my logic produces a clearly-wrong result. Here is my incorrect reasoning:
We can tackle this by counting the number of decks that satisfy the requirements. This should be:
(51 positions in the deck where a match can occur)(52 choices for the first card in the match)(2 choices for the second card in the match)(50 possibilities for the first other spot in the deck)(49 possibilities for the second other spot) . . . (2 possibilities for the next-to-last other spot in the deck)(1 possibility for the last spot in the deck) = 2 * 52!
However, there are only 52! shuffled decks possible, so there cannot be more that satisfy the requirements. What have I done wrong?
(Also, a Monte Carlo simulation for this question gives about 87% chance of having such a deck.)
Thanks in advance!