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For n \in \mathbb{N}, evaluate the determinant D = | n! (n+1)! (n+2)! | | (n+1)! (n+2)! (n+3)! | | (n+2)! (n+3)! (n+4)! |.

1. (n!)^2 (2n^3 − 8n^2)
2. (2n!)^3 (3n^2 + 4n − 5)
3. (n!)^3 (2n^3 + 8n^2 + 10n + 4)
4. None of these

Correct answer: (2n!)^3 (3n^2 + 4n − 5)

Solution

The correct option is derived from the properties of determinants and factorials, where the structure of the matrix leads to a specific polynomial expression in terms of n. By applying row operations and recognizing patterns in the factorial growth, we arrive at the determinant's value, confirming that it matches the form given in option B.

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