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Circular Permutations

  1. (i) Arrangements round a circular table
     
     The number of circular permutations of n different things taken all at a time is Description: 880.png(n – 1) !, if clockwise and anticlockwise orders are taken as different.
  2. Arrangements of beads or flowers (all different) around a circular necklace or garland:
     
    Thus the number of circular permutations of ‘n’ different things taken.
     
    All at a time is Description: 885.png(n – 1)!, if clockwise and anticlockwise orders are taken to be same.
Conditional Permutations
  1. Number of permutations of n things taking r at a time, in which a particular thing always occurs =Description: 890.png.
  2. Number of permutations of n things taking r at a time, in which a particular thing never occurs = Description: 896.png.
  3. Number of permutations of n different things taking all at a time, in which m specified things always come together =Description: 901.png.
  4. Number of permutations of n different things taking all at a time, in which m specified things never comes together = Description: 906.png 
  5. The number of permutations of 'n' things taken all at a time, when 'p' are alike of one kind, 'q' are alike of second, 'r' alike of third, and so onDescription: 911.png.
  6. The number of permutations of 'n' different things, taking 'r' at a time, when each thing can be repeated 'r' times = nr




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