The average (mean) score of a group of values is calculated by summing up all the values and dividing by the total number of values. In this case, the average score is given as 3.5.

Using the information provided in the table:

\(\begin{array}{c|c} \text{Scores} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text{No. of students} & 1 & 4 & 5 & 6 & x & 2 \end{array}\)

We can calculate the sum of scores:

\(1 \cdot 1 + 2 \cdot 4 + 3 \cdot 5 + 4 \cdot 6 + 5 \cdot x + 6 \cdot 2\)

Simplifying:

\(1 + 8 + 15 + 24 + 5x + 12\)

Now, set up the equation for the average score:

\(\frac{1 + 8 + 15 + 24 + 5x + 12}{1 + 4 + 5 + 6 + x + 2} = 3.5\)

Simplify the equation:

\(\frac{60 + 5x}{18 + x} = 3.5\)

Now, cross-multiply:

\(60 + 5x = 3.5(18 + x)\)

Simplify and solve for \(x\):

\(60 + 5x = 63 + 3.5x\)

\(5x - 3.5x = 63 - 60\)

\(1.5x = 3\)

\(x = \frac{3}{1.5} = 2\)

So, the value of x is 2