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 us analyze the problem using the principles of set theory and Venn diagrams for three sets: A, B, and C.
Let:
- $N$ be the total number of students, so $N=40$.
- $n\left(A\right)$ be the number of students who watched film A, so $n\left(A\right)=13$.
- $n\left(B\right)$ be the number of students who watched film B, so $n\left(B\right)=16$.
- $n\left(C\right)$ be the number of students who watched film C, so $n\left(C\right)=19$.
- $x$ be the number of students who watched all three films, which is represented by the intersection $A\cap B\cap C$.

According to the problem, each of the 40 students watched either only one film or all three films. This means:
1. There are no students who watched exactly two films.
2. There are no students who watched zero films (since the 40 students watched the films A, B, and C, and the categories partition the entire group of 40 students).
Thus, the total number of students is the sum of those who watched only one film and those who watched all three films.

Let us write the expressions for the number of students who watched only film A, only film B, and only film C:
- Since no students watched exactly two films, any student who watched film A either watched only A or watched all three (A, B, and C).
- Therefore, the number of students who watched only film A is:
$n\left(\text{only A}\right)=n\left(A\right)-x=13-x$
- Similarly, the number of students who watched only film B is:
$n\left(\text{only B}\right)=n\left(B\right)-x=16-x$
- And the number of students who watched only film C is:
$n\left(\text{only C}\right)=n\left(C\right)-x=19-x$

The total number of students $N$ is the sum of the students in all mutually exclusive regions of the Venn diagram:
$N=n\left(\text{only A}\right)+n\left(\text{only B}\right)+n\left(\text{only C}\right)+x$

Substitute the values into the equation:
$40=(13-x)+(16-x)+(19-x)+x$

Simplify the right-hand side of the equation:
$40=13+16+19-3x+x$
$40=48-2x$

Solve for $x$:
$2x=48-40$
$2x=8$
$x=4$

Thus, the number of students who watched all three films is 4.