The given problem states that $S = \{ X, 1, 7 \}$, where the mean of $S$ is also the median of $S$. We are tasked with finding the product of all possible values of $X$.

Step-by-step solution:

1. Mean of Set $S$: The mean (average) of a set is the sum of its elements divided by the number of elements. For the set $S = \{ X, 1, 7 \}$, the mean is: $\text{Mean of } S = \frac{X + 1 + 7}{3} = \frac{X + 8}{3}$

2. Median of Set $S$: To find the median, we need to arrange the elements in increasing order:

   • If $X \leq 1$, then the order is $\{ X, 1, 7 \}$, and the median is 1.
   • If $1 \leq X \leq 7$, then the order is $\{ 1, X, 7 \}$, and the median is $X$.
   • If $X \geq 7$, then the order is $\{ 1, 7, X \}$, and the median is 7.

3. Condition: The Mean Equals the Median We need to find values of $X$ such that the mean is equal to the median in each case.

   • Case 1: $X \leq 1$
     In this case, the median is 1. Setting the mean equal to the median: $\frac{X + 8}{3} = 1$ Solving for $X$: $X + 8 = 3 \quad \Rightarrow \quad X = -5$

   • Case 2: $1 \leq X \leq 7$
     In this case, the median is $X$. Setting the mean equal to the median: $\frac{X + 8}{3} = X$ Solving for $X$: $X + 8 = 3X \quad \Rightarrow \quad 8 = 2X \quad \Rightarrow \quad X = 4$

   • Case 3: $X \geq 7$
     In this case, the median is 7. Setting the mean equal to the median: $\frac{X + 8}{3} = 7$ Solving for $X$: $X + 8 = 21 \quad \Rightarrow \quad X = 13$

4. Possible Values of $X$: The possible values of $X$ that satisfy the condition are $X = -5$, $X = 4$, and $X = 13$.

5. Product of All Possible Values of $X$: The product of all possible values of $X$ is: $(-5) \times 4 \times 13 = -260$

Final Answer:

The product of all possible values of $X$ is $\boxed{-260}$.

Would you like further details on any part of this solution, or have additional questions? Here are some related questions for practice:

1. How do you calculate the mean of a set of numbers?
2. What is the difference between the mean and median of a dataset?
3. How would the solution change if $S$ had four elements instead of three?
4. What are the general steps to find the median of a set?
5. How does changing the value of $X$ influence the median?

Tip: When solving problems that involve both mean and median, carefully check how the median changes depending on the value of each element!