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 \(362880\).