variable x represents the base in which the number n can be expressed as $\overline{CBA}_{11}$.

Let's break down the information given:
In base 7, the number n can be expressed as $\overline{ABC}_7$.
In base x, the number n can be expressed as $\overline{CBA}_{11}$.
The largest possible value of n in base 10 is 247.

To find the value of x, we need to convert the given expressions into base 10.

In base 7, the number $\overline{ABC}_7$ can be expressed as $A \cdot 7^2 + B \cdot 7^1 + C \cdot 7^0$.

In base x, the number $\overline{CBA}_{11}$ can be expressed as $C \cdot x^2 + B \cdot x^1 + A \cdot x^0$.

According to the given information, the value of n is the same in both bases. Therefore, we can equate the expressions:

$A \cdot 7^2 + B \cdot 7^1 + C \cdot 7^0 = C \cdot x^2 + B \cdot x^1 + A \cdot x^0$

To find the largest possible value of n, we need to find the largest possible values for A, B, and C in base 7.

Since A, B, and C are digits in base 7, they can take values from 0 to 6.

Let's substitute the largest possible values for A, B, and C into the equation:

$6 \cdot 7^2 + 6 \cdot 7^1 + 6 \cdot 7^0 = 6 \cdot x^2 + 6 \cdot x^1 + 6 \cdot x^0$

Simplifying the equation:

294 + 42 + 6 = 6x^2 + 6x + 6

342 = 6x^2 + 6x + 6

Dividing both sides of the equation by 6:

57 = x^2 + x + 1

Now, we need to find the largest possible value of x that satisfies this equation.

By trial and error, we find that x = 11 is the largest possible value that satisfies the equation.

Therefore, the value of the unknown variable x is 11.
The answer is: 11
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Suppose that $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\overline{ABC}_7$, and in base X$, then $n$ can be expressed as $\overline{CBA}_{11}$. Find the largest possible value of $n$ in base $10$. The answer is 247. What is the value of unknown variable X?

### Answer:
