For some natural number $$ n $$, assume that $$ (15,000)! $$ is divisible by $$ (n!)! $$. The largest possible value of $$ n $$ is:

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Accepted Answer

1. **Problem Setup**:  
   We need to find the largest natural number $$ n $$ such that:  
   $$
   (n!)! \leq (15,000)!
   $$

2. **Evaluate Factorial Values**:  
   Start calculating factorials for small values of $$ n $$:  
   - For $$ n = 5 $$:  
     $$
     5! = 120.
     $$
     $$
     (5!)! = 120! 	ext{ (well within } 15,000! 	ext{)}.
     $$

- For $$ n = 6 $$:  
     $$
     6! = 720.
     $$
     $$
     (6!)! = 720! 	ext{ (still within } 15,000! 	ext{)}.
     $$

- For $$ n = 7 $$:  
     $$
     7! = 5,040.
     $$
     $$
     (7!)! = 5,040! 	ext{ (still within } 15,000! 	ext{)}.
     $$

- For $$ n = 8 $$:  
     $$
     8! = 40,320.
     $$
     $$
     (8!)! > (15,000)! 	ext{ (exceeds the limit)}.
     $$

3. **Conclusion**:  
   The largest value of $$ n $$ such that $$ (n!)! \leq (15,000)! $$ is **$$ n = 7 $$**.

Thus, the correct answer is **B. 7**.

Answer

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