Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?

The correct option is 4.

Let's break down the problem step-by-step using set theory and algebra.

We are given that there are 40 students in total, and each student watched either only one film (A, B, or C) or all three films. None of the students watched exactly two films.

Let us define the variables for the number of students in each category:
- Let $x$ be the number of students who watched only film A.
- Let $y$ be the number of students who watched only film B.
- Let $z$ be the number of students who watched only film C.
- Let $w$ be the number of students who watched all three films (A, B, and C).

Since the total number of students is 40, we can write our first equation as:

$$x+y+z+w=40$$

Next, we are given the total number of students who watched each individual film. A student who watched film A must have either watched only film A or all three films. Therefore, we can write the equations for each film as follows:

For film A:
$$x+w=13$$

For film B:
$$y+w=16$$

For film C:
$$z+w=19$$

Now, we can add these three individual film equations together:

$$(x+w)+(y+w)+(z+w)=13+16+19$$

Simplifying this, we get:

$$x+y+z+3w=48$$

We can rewrite this expression to separate the total student count term:

$$(x+y+z+w)+2w=48$$

Substitute the first equation ($x+y+z+w=40$) into the equation:

$$40+2w=48$$

Subtract 40 from both sides:

$$2w=8$$

Divide by 2:

$$w=4$$

Thus, exactly 4 students watched all three films.