How many anagrams are there for the word "MAGENTA" such that only one vowel appears in its original position?
(a) $1512$
(b) $1152$
(c) $1008$
(d) $720$
(e) $480$
According to the official key, the answer is $1152$. I'm familiar with the concept of a derangement, that the derangement of $n$ objects can be calculated as:
$$D_n=n!\cdot \sum_{k=0}^{n}\frac{(-1)^k}{k!}.$$
But I couldn't associate it with the fact that there's only one letter on its original position.
Best Answer
Strategy: Use the generalized Inclusion-Exclusion Principle. Include those arrangements with at least one vowel in its original position, subtract those arrangements with at least two vowels in their original positions, then add those arrangements with all three vowels in their original positions.
At least one vowel in its original position:
At least two vowels in their original positions:
All three vowels in their original positions: