Find the missing number in the grid: [[7, 4, 5], [8, 7, 6], [3, 3, ?], [29, 19, 31]]

The grid is structured as follows:

$$\begin{bmatrix} 7 & 4 & 5 \\ 8 & 7 & 6 \\ 3 & 3 & ? \\ 29 & 19 & 31 \end{bmatrix}$$

Let's examine the relationship between the numbers in each column.

For the first column, the numbers are 7, 8, 3, and 29.
Let's try the operation: (first number $\times$ third number) + second number.

$$(7 \times 3) + 8 = 21 + 8 = 29$$

This matches the fourth number in the first column.

For the second column, the numbers are 4, 7, 3, and 19.
Applying the same pattern:

$$(4 \times 3) + 7 = 12 + 7 = 19$$

This also matches the fourth number in the second column, confirming the pattern.

For the third column, the numbers are 5, 6, ?, and 31. Let the missing number be $x$.
Using the established pattern:

$$(5 \times x) + 6 = 31$$

$$5x + 6 = 31$$

Subtract 6 from both sides:

$$5x = 31 - 6$$ $$5x = 25$$

Divide by 5:

$$x = \frac{25}{5}$$ $$x = 5$$

The missing number is 5.

The options are: a) 3, b) 4, c) 5, d) 6.
Our calculated missing number is 5, which corresponds to option c.

The final answer is $\text{c) 5}$.