+88 Four people sit around a circular table, and each person will roll a standard six-sided die. What is the probability that no two people sitting next to each other will roll the same number after they each roll the die once? Express your answer as a common fraction.
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+6251 Player 1 rolls a die. There are 6 possible rolls.
Player 2 then has to roll a different number. There are 5 possible rolls.
Players 3 and 4 then have to roll such that neither they nor player 4 and 1 have the same roll.
There are 4 rolls for player 3 that lead to 4 possible rolls for player 4 and
There is 1 possible roll (the value that player 1 rolled) that leads to 5 possible rolls for player 4.
Combining all this we have that there are
6 * 5 * (4 * 4 + 1 * 5) = 30*21 = 630 valid rolls among the players.
There are a total of 64 = 1296 possible rolls among the players.
Thus the probability of no consecutive rolls in the ring being the same is
\(p = \dfrac{630}{1296}=\dfrac{35}{72}\)
