Shuffled Deck
You have 52 cards. The cards are all shuffled and are rotated on their faces as well as their horizontal axis. (e.g. You pull out a card, it turns out to be a Queen of Hearts but may or may not be face up or face down, and/or may or may not be rotated clockwise or counter-clockwise 180 degrees) You shuffle all the cards. What are the total no. of possibilities of the shuffled deck?
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2 solutions
Total No. of Cards = 5 2
Possibilities for 1st card = 5 2 ∗ 4 [There are 4 ways a card could appear: FaceUp+Not Rotated, FaceDown+Not Rotated, FaceUp + Rotated, FaceDown + Rotated, and there are 52 card possibilities for the first card]
Possibilities for 2nd card = 5 1 ∗ 4 [as, there would be one less card in the deck due to that card already being selected as the first card, so one less possibility of the card itself, but the way this card is kept, can also vary]
...
Possibilities for the 52nd Card = 1 ∗ 4 [51 cards except 1 would already be selected some way or the other, so that leaves only 1 card possibility for the last card]
So, total possibilities are: 5 2 ∗ 4 ∗ 5 1 ∗ 4 ∗ 5 0 ∗ 4 ∗ . . . ∗ 1 ∗ 4 Rearranging the terms, we get: [ 5 2 ∗ 5 1 ∗ 5 0 ∗ . . . ∗ 1 ] ∗ [ 4 1 ∗ 4 2 ∗ 4 3 ∗ 4 4 ∗ . . . ∗ 4 5 2 ] Which equals: [ 5 2 ! ] ∗ [ 4 5 2 ]
Also note: This is a huuuuuge no. of possibilities, so much so, that it equals: 1 . 6 3 5 9 4 2 1 4 6 7 0 0 1 7 0 8 3 6 5 2 4 3 7 6 2 7 9 4 4 9 9 e + 9 9
There are 5 2 ! ways the cards can be ordered, and 4 ways to orient each card.
So,
N = 4 5 2 ⋅ 5 2 !