Step-by-Step Solution using Set Theory

This problem can be solved by setting up and solving a system of equations based on the given information. A key piece of information is that each student watched either exactly one film or all three films; no student watched exactly two.

1. Define the Variables

• Let A be the number of students who watched only film A.
• Let B be the number of students who watched only film B.
• Let C be the number of students who watched only film C.
• Let x be the number of students who watched all three films.

2. Formulate the Equations

We can create equations based on the totals given:

1. The total number of students is the sum of all the groups:
   A + B + C + x = 40
2. The total who watched film A includes those who watched only A and those who watched all three:
   A + x = 13  →  A = 13 - x
3. The total who watched film B includes those who watched only B and those who watched all three:
   B + x = 16  →  B = 16 - x
4. The total who watched film C includes those who watched only C and those who watched all three:
   C + x = 19  →  C = 19 - x

3. Solve for x

Now we substitute the expressions for A, B, and C from equations (2), (3), and (4) into the main equation (1):

(13 - x) + (16 - x) + (19 - x) + x = 40

Combine the constant terms and the 'x' terms:

(13 + 16 + 19) + (-x - x - x + x) = 40

48 - 2x = 40

Subtract 40 from both sides:

48 - 40 = 2x

8 = 2x

x = 4

Conclusion

The number of students who watched all three films is 4.