Question
Six dancers are to be lined up for a stage entry. Among
them, three are from the same team and must not be adjacent. How many valid line-ups are possible?Solution
6 dancers can be arranged among themselves in 6P6 ways = 720 ways.
Assume that the 3 team dancers are one entity. The total number of ways in which they can be arranged among themselves = 3! = 6 ways.
Also, the set of three team dancers and the other students can be arranged among themselves = 4! = 24 ways.
Thus, total number of ways in which three dancers are together = 6 × 24 = 144
Thus, number of ways in which the 3 dancers will not stand together = 720 − 144 = 576
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