The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ______.

Correct Answer:

12

Solution:

Given L I L A C

Step 1: L's are indistinguishable, by fixing their positions, we get:

  • Case 1:    L    L   
  • Case 2:    L     L
  • Case 3:          L L

no character should appear in its original position

Step 2: Arranging the viewing letters: I, A, C

  • Case 1:      L    L   
    Letter C can take only 2 places.
    Letter A and I can take any of the 2 remaining places.
    = 2 × 2 × 1 = 4
  • Case 2:     L     L
    Letter A can take only 2 places.
    Letter C and I can take any of the 2 remaining places.
    = 2 × 2 × 1 = 4
  • Case 3:         L L
    Letter I can take only 2 places.
    Letter A and C can take any of the 2 remaining places.
    = 2 × 2 × 1 = 4

Total Ways = Case 1 + Case 2 + Case 3
= 4 + 4 + 4 = 12 Ways