How many anagrams are there for the word “MAGENTA” such that only one vowel appears in its original position

combinatoricscombinatorics-on-words

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:

  1. Include those arrangements with the first A in its original position, with the E in its original position, and with the second A in its original position.
  2. Exclude those arrangements with the first A and the E in their original positions, with the two A's in their original positions, and with the E and second A in their original positions.
  3. Include those arrangements with all three vowels in their original positions.

At least two vowels in their original positions:

  1. Include those arrangements with the first A and the E in their original positions, with the two A's in their original positions, and with the E and second A in their original positions.
  2. Exclude those arrangements with all three vowels in their original positions.

All three vowels in their original positions:

  1. Include those arrangements with all three vowels in their original positions.
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