For some natural number $n$ , assume that $(15000)!$ is divisible by $(n!)!$ . The largest possible value of $n$ is

$5$

$7$

$6$

$4$

✅ Correct Option: 2

We need to find the largest natural number $n$ such that $(15000)!$ is divisible by $(n!)!$ .

For $(15000)!$ to be divisible by $(n!)!$ , we need:

$(n!)! \leq 15000!$

This fundamental rule of factorials tells us that $a!$ divides $b!$ if and only if $a \leq b$ .

Therefore, our condition becomes:

$n! \leq 15000$

Now we need to find the largest $n$ where $n! \leq 15000$ .

Let us calculate factorials:

$1! = 1$

$2! = 2$

$3! = 6$

$4! = 24$

$5! = 120$

$6! = 720$

$7! = 5040$ (This is $\leq 15000$ )

$8! = 40320$ (This exceeds 15000)

Since $7! = 5040 \leq 15000$ but $8! = 40320 > 15000$ , the largest possible value of $n$ is $7$ .

When $n = 7$ : $(n!)! = (7!)! = 5040!$

Since $5040 \leq 15000$ , we have $5040! \leq 15000!$

When $n = 8$ : $(n!)! = (8!)! = 40320!$

Since $40320 > 15000$ , we cannot have $40320! \leq 15000!$

The factorial function grows extremely rapidly. This is why even a small increase from $n = 7$ to $n = 8$ makes such a dramatic difference (from $5040$ to $40320$ ).

Therefore, the largest possible value of $n$ is $\boxed{7}$ .

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