This problem involves understanding the distribution of students watching films based on specific conditions. We are given the total number of students and the number of students who watched individual films, with a key condition that each student either watched only one film or all three films. Our goal is to determine how many students watched all three films.

Understanding the Film Viewing Pattern

The question states a very important condition: "Each student watched either only one film or all three." This simplifies the problem significantly because it means there are no students who watched exactly two films (e.g., only A and B, or only B and C, or only A and C).

• Let \(N_{Total}\) be the total number of students.
• Let \(N_A\) be the number of students who watched film A.
• Let \(N_B\) be the number of students who watched film B.
• Let \(N_C\) be the number of students who watched film C.
• Let \(N_{OnlyA}\) be the number of students who watched only film A.
• Let \(N_{OnlyB}\) be the number of students who watched only film B.
• Let \(N_{OnlyC}\) be the number of students who watched only film C.
• Let \(N_{AllThree}\) be the number of students who watched all three films (A, B, and C).

Given Film Data

We are provided with the following information:

• Total students: \(N_{Total} = 40\)
• Students who watched film A: \(N_A = 13\)
• Students who watched film B: \(N_B = 16\)
• Students who watched film C: \(N_C = 19\)

The crucial condition "Each student watched either only one film or all three" implies that:

• If a student watched film A, they either watched only film A or all three films. So, \(N_A = N_{OnlyA} + N_{AllThree}\).
• Similarly, \(N_B = N_{OnlyB} + N_{AllThree}\).
• And \(N_C = N_{OnlyC} + N_{AllThree}\).

Also, the total number of students can be expressed as the sum of those who watched only one film and those who watched all three films:

\(N_{Total} = N_{OnlyA} + N_{OnlyB} + N_{OnlyC} + N_{AllThree}\)

Calculating Students Who Watched All Three Films

Let's use the relationships derived from the given conditions to find \(N_{AllThree}\).

From the individual film counts, we can express the "only" categories in terms of \(N_{AllThree}\):

• \(N_{OnlyA} = N_A - N_{AllThree} = 13 - N_{AllThree}\)
• \(N_{OnlyB} = N_B - N_{AllThree} = 16 - N_{AllThree}\)
• \(N_{OnlyC} = N_C - N_{AllThree} = 19 - N_{AllThree}\)

Now, substitute these into the equation for total students:

\(N_{Total} = N_{OnlyA} + N_{OnlyB} + N_{OnlyC} + N_{AllThree}\)

\(40 = (13 - N_{AllThree}) + (16 - N_{AllThree}) + (19 - N_{AllThree}) + N_{AllThree}\)

Combine the constant terms and the \(N_{AllThree}\) terms:

\(40 = (13 + 16 + 19) + (-N_{AllThree} - N_{AllThree} - N_{AllThree} + N_{AllThree})\)

\(40 = 48 + (-3 N_{AllThree} + N_{AllThree})\)

\(40 = 48 - 2 N_{AllThree}\)

Now, solve for \(N_{AllThree}\):

\(2 N_{AllThree} = 48 - 40\)

\(2 N_{AllThree} = 8\)

\(N_{AllThree} = \frac{8}{2}\)

\(N_{AllThree} = 4\)

Verification of Film Counts

Let's verify our result by calculating the number of students who watched only one film:

• Students who watched only film A: \(N_{OnlyA} = 13 - 4 = 9\)
• Students who watched only film B: \(N_{OnlyB} = 16 - 4 = 12\)
• Students who watched only film C: \(N_{OnlyC} = 19 - 4 = 15\)

Total students = (Only A) + (Only B) + (Only C) + (All Three)

\(Total = 9 + 12 + 15 + 4\)

\(Total = 40\)

This matches the total number of students given in the question, confirming our calculation is correct.

Therefore, 4 students watched all three films.