Solving for the Factorial Value $(3x)!$

This explanation details the steps to find the value of $(3x)!$ using the provided information about factorials and an equation.

Understanding Factorial Notation

The factorial of a non-negative integer $n$, denoted as $n!$, represents the product of all positive integers up to $n$. For instance, $4! = 4 \times 3 \times 2 \times 1 = 24$. The question involves $x!$ and $(2x)!$.

Analyzing the Given Equation

We are given the following:

• $y = x!$
• $z = (2x)!$
• The relationship $\frac{z}{y} = 120$

Let's substitute the expressions for $y$ and $z$ into the given relationship:

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

Determining the Value of x

To solve this factorial equation, we can expand $(2x)!$. Recall that $(2x)! = (2x) \times (2x-1) \times \dots \times (x+1) \times x!$. Substituting this into the equation:

$ \frac{(2x) \times (2x-1) \times \dots \times (x+1) \times x!}{x!} = 120 $

Cancel out the $x!$ term:

$ (2x) \times (2x-1) \times \dots \times (x+1) = 120 $

This equation shows the product of $x$ consecutive integers starting from $2x$ down to $x+1$. We can find the value of $x$ by testing small positive integers:

• If $x=1$: $\frac{(2 \times 1)!}{1!} = \frac{2!}{1!} = \frac{2}{1} = 2$. This does not equal $120$.
• If $x=2$: $\frac{(2 \times 2)!}{2!} = \frac{4!}{2!} = \frac{24}{2} = 12$. This does not equal $120$.
• If $x=3$: $\frac{(2 \times 3)!}{3!} = \frac{6!}{3!} = \frac{720}{6} = 120$. This matches the given condition.

So, the value of $x$ that satisfies the equation is $3$. We successfully found the unknown value $x$.

Calculating the Final Answer $(3x)!$

The question asks for the value of $(3x)!$. Since we found $x=3$, we substitute this value:

$ (3x)! = (3 \times 3)! = 9! $

Now, we compute the value of $9!$:

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

Let's calculate the product step-by-step:

• $9 \times 8 = 72$
• $72 \times 7 = 504$
• $504 \times 6 = 3024$
• $3024 \times 5 = 15120$
• $15120 \times 4 = 60480$
• $60480 \times 3 = 181440$
• $181440 \times 2 = 362880$
• $362880 \times 1 = 362880$

Therefore, the value of $(3x)!$ is $362880$.