The size of the pool of possible cheat codes = 46233.
Let us try to enter a cheat code into a video game.
You know that in order to enter the cheat code you must input more than one and up to eight button presses before the pressing the start button.
Given that:
Total buttons - 9 buttons
Input button - more than one and up to 8 presses
1! to 8! = 1 + 2 + 6 + 24 + 120 + 720 + 5040 + 40320
= 46233 cheat codes.
Size of the pool of possible cheat codes = 46233 cheat codes.
Hence the answer is the size of the pool of possible cheat codes = 46233.
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