Question

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 ___

• A. 12

Answer 1

The correct option is A 12
Since both L's are indistinguishable. First L's can be arranged in 3 positions 2, 3, or 5 in ${}^3$$C_2$ = 3 ways as follows:
_ $\underset{–}{L}$ _ $\underset{–}{L}$ _
or _ $\underset{–}{L}$ _ _ $\underset{–}{L}$
or _ _ _ $\underset{–}{L}$ $\underset{–}{L}$
Now the letters I, A, C can be deranged in 2 $\times$ 2! ways. Example in _ $\underset{–}{L}$ _$\underset{–}{L}\_$ .C cannot occupy 5th position, so only 2 ways. Remaining I and A can be arranged in remaining 2 position in 2! ways = 2 ways.
So answer is 3 x 2 x 2! = 12.