Student Film Watching Problem Setup
We are given the total number of students and information about the films they watched. The core condition is that each student watched either exactly one film or all three films. Our objective is to determine the number of students who watched all three films.
Variables and Equations for Film Watching
Let $N$ represent the total number of students, where $N = 40$. Let $A_{only}$, $B_{only}$, and $C_{only}$ denote the number of students who watched only film A, only film B, and only film C, respectively. Let $X$ represent the number of students who watched all three films (A, B, and C).
According to the problem statement, students fall into two categories: those watching only one film and those watching all three. Thus, the total number of students can be expressed as:
$ N = A_{only} + B_{only} + C_{only} + X $
We are provided with the total counts for each film's viewership:
- Total who watched A = 13. This group comprises those who watched only A and those who watched all three: $A_{only} + X = 13$.
- Total who watched B = 16. This group comprises those who watched only B and those who watched all three: $B_{only} + X = 16$.
- Total who watched C = 19. This group comprises those who watched only C and those who watched all three: $C_{only} + X = 19$.
From these equations, we can express the counts of students watching only a single film in terms of $X$:
- $A_{only} = 13 - X$
- $B_{only} = 16 - X$
- $C_{only} = 19 - X$
Calculating Students Watching All Three Films
Substitute these expressions into the equation for the total number of students:
$ 40 = (13 - X) + (16 - X) + (19 - X) + X $
Combine the constant terms and the $X$ terms:
$ 40 = (13 + 16 + 19) + (-X - X - X + X) $
$ 40 = 48 - 2X $
Now, solve for $X$:
$ 2X = 48 - 40 $
$ 2X = 8 $
$ X = \frac{8}{2} $
$ X = 4 $
Thus, 4 students watched all three films.
Verification of Film Watching Numbers
To confirm our result, let's calculate the number of students in each category:
- Students who watched only A: $A_{only} = 13 - 4 = 9$
- Students who watched only B: $B_{only} = 16 - 4 = 12$
- Students who watched only C: $C_{only} = 19 - 4 = 15$
- Students who watched all three films: $X = 4$
The sum is $9 + 12 + 15 + 4 = 40$, which matches the total number of students provided in the question.
Important Questions from Venn Diagrams
Each row of Column-I has three items and each item is represented by a circle in Column-II. The arrangement of circles in Column-II represents the relationship among the items in Column-I.
Identify the option that has the most appropriate match between Column-I and Column-II.
Note: The figure shown are representative.
Column-I Column-II (1) Animal, Zebra, Giraffe (P) 
(2) Director, Producer, Actor (Q) 
(3) Word, Sentence, Novel (R) 
(4) Pianist, Guitarist, Instrumentalist (S) 
- To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?
- 500 students are taking one or more courses out of Chemistry, Physics, and Mathematics. Registration records indicate course enrolment as follows: Chemistry (329), Physics (186), Mathematics (295), Chemistry and Physics (83), Chemistry and Mathematics (217), and Physics and Mathematics (63). How many students are taking all 3 subjects?
In the given diagram, teachers are represented in the triangle, researchers in the circle and administrators in the rectangle. Out of the total number of the people, the percentage of administrators shall be in the range of ___________.
Out of \(100\) textile companies, \(10\) companies are involved in spinning, weaving and chemical processing, \(25\) companies are involved in spinning and chemical processing, and \(30\) companies are involved in weaving and chemical processing. If \(65\) companies are involved in chemical processing, the number of companies involved {ONLY} in chemical processing is ________________________.
