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      Question

      Five singers are to be arranged in a line for a concert

      photo. Two of them are siblings and must not stand next to each other. How many valid arrangements are possible?
      A 96 Correct Answer Incorrect Answer
      B 84 Correct Answer Incorrect Answer
      C 72 Correct Answer Incorrect Answer
      D 64 Correct Answer Incorrect Answer
      E 60 Correct Answer Incorrect Answer

      Solution

      5 singers can be arranged among themselves in 5P5 ways = 120 ways.
      Assume that the 2 siblings are one entity. The total number of ways in which they can be arranged among themselves = 2! = 2 ways.
      Also, the set of two siblings and the other students can be arranged among themselves = 4! = 24 ways.
      Thus, total number of ways in which two siblings are together = 2 × 24 = 48
      Thus, number of ways in which the siblings will not stand together = 120 − 48 = 72

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