Let y = x! and z = (2x)! . If (z/y) = 120 , then what is the value...

Given $y = x!$ and $z = (2x)!$.

We are given $\dfrac{z}{y} = 120$.

Substituting the values of $y$ and $z$, we get:

$\dfrac{(2x)!}{x!} = 120$

Checking for small integer values of $x$:

For $x = 1$, $\dfrac{2!}{1!} = 2$

For $x = 2$, $\dfrac{4!}{2!} = \dfrac{24}{2} = 12$

For $x = 3$, $\dfrac{6!}{3!} = \dfrac{720}{6} = 120$

Thus, $x = 3$.

We need to find the value of $(3x)!$.

Substituting $x = 3$, we get $(3 \times 3)! = 9!$.

$9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880$.

Answer: $362880$