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Shortcut 95: Repeated elements      n!/(a!*b!) Question: In how many ways the letters of the word “ENVIRONMENT” can be arranged? Answer: n = 11 Let, a = 2 (E is repeated twice) Let, b = 3 (N is repeated thrice) Number of arrangements = 11!/(2! x 3!) Shortcut 96: Circular arrangement   (n-1)! Question: In how many ways 6 persons can be arranged in a circle? Answer: n = 6 Number of arrangements = (6 – 1)! = 5! = 120 Shortcut 97: Circular arrangement with elements occurring together   2!*(n-2)!        3!*(n-3)! Question: In how many ways 8 persons can be seated around a circular table with two persons always sitting together? Answer: 2! x (8 – 2)! = 2 x 720 = 1440 Note: If three persons are sitting together, then 3! x (8 – 3)! If four persons are sitting together, then 4! x (8 – 4)! Shortcut 98: Arranging a necklace with beads (n-1)!/2 Question: In how many ways a necklace with 8 beads can be arranged? Answer: n = 8 Number of arrangements = (8 – 1)!/2 = 2520]]>

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