The number of permutations of the characters in LILAC so that no character…

Answer: 12

Reasoning: The word LILAC has letters at positions 1:L, 2:I, 3:L, 4:A, 5:C with the two L's indistinguishable. A valid arrangement must place no letter in its original position. Since L cannot occupy positions 1 or 3, both L's must occupy two of the positions 2, 4, 5. The remaining three positions will be filled by I, A, and C. For each choice of which single position among 2, 4, 5 is NOT taken by an L, count the valid permutations of I, A, C into the three remaining positions avoiding their original spots.

• If the non-L position is 2 (so L's occupy positions 4 and 5): place I, A, C into positions 1, 2, 3. Total permutations = 3! = 6. Exclude those with I at position 2 (2 permutations), leaving 4 valid arrangements.

• If the non-L position is 4 (so L's occupy positions 2 and 5): place I, A, C into positions 1, 3, 4. Total permutations = 6. Exclude those with A at position 4 (2 permutations), leaving 4 valid arrangements.

• If the non-L position is 5 (so L's occupy positions 2 and 4): place I, A, C into positions 1, 3, 5. Total permutations = 6. Exclude those with C at position 5 (2 permutations), leaving 4 valid arrangements.

Adding the valid arrangements from all three cases gives 4 + 4 + 4 = 12, which is the required number of permutations.