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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