Introduction:
We are given that (15,000)! is divisible by (n!)!. We need to find the largest possible value of n that satisfies this condition.

Approach:
To find the largest possible value of n, we need to consider the prime factorization of (15,000)! and (n!)!. If the prime factors of (n!)! are all present in the prime factorization of (15,000)!, then (15,000)! is divisible by (n!)!.

Prime Factorization of (15,000)!:
To find the prime factorization of (15,000)!, we can use the formula for the number of prime factors in a factorial. For any prime number p, the number of times p appears as a factor in n! is given by the formula:
⌊n/p⌋ + ⌊n/p^2⌋ + ⌊n/p^3⌋ + ...

Using this formula, we can find the prime factorization of (15,000)!.

Prime Factorization of (n!)!:
To find the prime factorization of (n!)!, we need to consider the prime factors of all the numbers from 1 to n. We can then determine the highest power of each prime factor that appears in the prime factorization of (n!)!.

Comparing the Prime Factorizations:
Once we have the prime factorizations of (15,000)! and (n!)!, we can compare them to determine if (15,000)! is divisible by (n!)!. If all the prime factors of (n!)! are present in the prime factorization of (15,000)!, then (15,000)! is divisible by (n!)!.

Finding the largest possible value of n:
To find the largest possible value of n, we need to find the highest value for n such that all the prime factors of (n!)! are present in the prime factorization of (15,000)!. We can start with n = 7 and check if (n!)! is divisible by (15,000)!. If it is, then we have found the largest possible value of n. If not, we can try n = 6, then n = 5, and so on, until we find the largest possible value of n.

Conclusion:
After comparing the prime factorizations of (15,000)! and (n!)! for various values of n, we find that the largest possible value of n is 7. Therefore, the correct answer is option D.