The question asks to determine the product of all possible values of $X$ in the set $S = \{X, 1, 7\}$, under the condition that the mean of $S$ is also the median of $S$.

Step 1: Definitions

• Mean: The mean of the set $S$ is given by: $\text{Mean} = \frac{X + 1 + 7}{3} = \frac{X + 8}{3}.$

• Median: The median of $S$ depends on the value of $X$. Since the set has three elements, the median is the middle value when $S$ is arranged in increasing order.

Step 2: Analyze the possible arrangements of $S$

1. Case 1: $X \leq 1$

   • The set $S$ in ascending order is $\{X, 1, 7\}$.
   • The median is $1$ (the middle value).

2. Case 2: $1 \leq X \leq 7$

   • The set $S$ in ascending order is $\{1, X, 7\}$.
   • The median is $X$ (the middle value).

3. Case 3: $X \geq 7$

   • The set $S$ in ascending order is $\{1, 7, X\}$.
   • The median is $7$ (the middle value).

Step 3: Equating the mean and the median

Case 1: $\text{Mean} = \text{Median} = 1$

$\frac{X + 8}{3} = 1 \implies X + 8 = 3 \implies X = -5.$

Case 2: $\text{Mean} = \text{Median} = X$

$\frac{X + 8}{3} = X \implies X + 8 = 3X \implies 8 = 2X \implies X = 4.$

Case 3: $\text{Mean} = \text{Median} = 7$

$\frac{X + 8}{3} = 7 \implies X + 8 = 21 \implies X = 13.$

Step 4: Validating the cases

• For $X = -5$, $S = \{-5, 1, 7\}$, and the mean and median are both $1$. Valid.
• For $X = 4$, $S = \{1, 4, 7\}$, and the mean and median are both $4$. Valid.
• For $X = 13$, $S = \{1, 7, 13\}$, and the mean and median are both $7$. Valid.

Step 5: Product of all possible $X$

The possible values of $X$ are $-5, 4, 13$. Their product is: $(-5) \cdot 4 \cdot 13 = -260.$

Final Answer:

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

Would you like a detailed breakdown of any step?

Here are 5 related questions for practice:

1. How is the median calculated for a set with an even number of elements?
2. What happens if $X$ is a fractional value? How does it affect the solution?
3. How would the problem change if the mean had to equal the mode instead of the median?
4. What are the conditions for a set to have equal mean, median, and mode?
5. How can this problem be generalized for larger sets with variable elements?

Tip: Always verify your solutions by substituting the results back into the original conditions.