Question
Seven people are to be arranged on seven chairs. Among
them, there are three friends who should not sit together. Find the number of valid arrangements.Solution
7 people can be arranged among themselves in 7P7 ways = 5040 ways.
Assume that the 3 friends are one entity. The total number of ways in which they can be arranged among themselves = 3! = 6 ways.
Also, the set of three friends and the other students can be arranged among themselves = 5! = 120 ways.
Thus, total number of ways in which three friends are together = 6 × 120 = 720
Thus, number of ways in which all 3 friends will not occupy consecutive seats = 5040 − 720 = 4320
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