Given word: LILAC
It is given that the two $L$'s are indistinguishable and need to calculate the number of permutations of the word such that no character in the permuted word appears in its original position.
The way I approached is to find the number of derangements of 5 objects (treating the two $L$'s as distinct) and then dividing the answer by $2!$ giving $\frac{44}{2} = 22$.
Some are claiming that the answer is $12$ instead of $22$.
Is $22$ the wrong answer ? If yes, then how and why ?
Thanks.
Best Answer
I don't understand the logic behind dividing by $2$. To do this via derangements, you'd need to exclude the derangements in which the second $L$ winds up in the first slot and/or the first $L$ winds up in the third slot.
To count by hand:
There are three choices for the first slot, and then two for the third slot.
That leaves one letter out of $I,A,C$ unchosen and there are two slots that one might occupy. After that, the $L's$ must go in the two unfilled slots.
Hence the answer is $$3\times 2\times 2 =12$$