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Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?


How many different $6$-digit numbers can be formed by using digits $1$, $2$, and $3$, if no adjacent digits can be the same?


How many ways are there to arrange $8$ girls and $25$ boys to sit around a table so that there are at least $2$ boys between any pair of girls? If a sitting plan can be simply rotated to match another one, these two are treated as the same.