Hello guys! I am pretty new to the community. In fact, what brought me here is this tricky problem I found and I can't solve it!
The problem is the following:
In one class the professor brought 2 decks of cards each containing 52 standard cards. The one deck was blue backed and the other was red backed. He then gave the decks to his assistant after which the assistant shuffled them nicely. After that, the professor left the room. The assistant then chose 5 students at random and let each one of them take one card and give it to him. The students did exactly that.Then the professor came back inside and the assistant revealed 4 of the cards to the professor in a previously arranged order (regarding their faces of the cards) to the professor (the professor can see the backs of the cards as well). Prove that taking into account the order of the revealed cards the professor can guess correctly the 5th card's color (red or black), value (A,2,3,4,5...J,Q,K) and back color (red or blue).
And also a sub-problem to this one:
If the professor had one deck of cards of one back color (for example red), and the story was all the same except that the students took 4 cards and the assistant revealed 3 of the cards to the professor in a previously arranged order (regarding their faces), prove that the professor can correctly guess the color and value of the 4th card!
Thank you very much in advance, I really hope this community can help me! :)