Given:
Total students [math]n(U)=100[/math]
Number of students who like to play football [math]n(F)=60[/math]
Number of students who like to play volleyball [math]n(V)=48[/math]
Number of students who like to play cricket [math]n(C)=40[/math]
Number of students who like to play football and volleyball [math]n(F \cap V)=32[/math]
Number of students who like to play football and cricket [math]n(F \cap C)=22[/math]
Number of students who like to play volleyball and cricket [math]n(V \cap C)= 20[/math]
Number of students who like to play all games [math]n(F \cap V \cap C) = 5[/math]

In a Venn diagram, find the number of students who like at least one game.

Use the Principle of Inclusion-Exclusion to find the number of students who like at least one game.
The formula is: n ( F o b re ak \tcup V o b re ak \tcup C ) = n ( F ) + n ( V ) + n ( C ) − n ( F o b re ak \tcap V ) − n ( F o b re ak \tcap C ) − n ( V o b re ak \tcap C ) + n ( F o b re ak \tcap V o b re ak \tcap C ) .
Substitute the given values into the formula: n ( F o b re ak \tcup V o b re ak \tcup C ) = 60 + 48 + 40 − 32 − 22 − 20 + 5 .
Calculate the result: n ( F o b re ak \tcup V o b re ak \tcup C ) = o b re ak \tboxed 79 .

Explanation

Understand the problem and provided data We are given the number of students who like football, volleyball, and cricket, as well as the number of students who like each pair of sports and all three sports. We want to find the number of students who like at least one of these sports. This is a classic application of the Principle of Inclusion-Exclusion.

State the Principle of Inclusion-Exclusion The Principle of Inclusion-Exclusion for three sets (in this case, the sets of students who like football, volleyball, and cricket) states that:


n ( F o b re ak \tcup V o b re ak \tcup C ) = n ( F ) + n ( V ) + n ( C ) − n ( F o b re ak \tcap V ) − n ( F o b re ak \tcap C ) − n ( V o b re ak \tcap C ) + n ( F o b re ak \tcap V o b re ak \tcap C )
Where:

n ( F o b re ak \tcup V o b re ak \tcup C ) is the number of students who like at least one of the three sports.
n ( F ) is the number of students who like football.
n ( V ) is the number of students who like volleyball.
n ( C ) is the number of students who like cricket.
n ( F o b re ak \tcap V ) is the number of students who like both football and volleyball.
n ( F o b re ak \tcap C ) is the number of students who like both football and cricket.
n ( V o b re ak \tcap C ) is the number of students who like both volleyball and cricket.
n ( F o b re ak \tcap V o b re ak \tcap C ) is the number of students who like all three sports.


List the given values We are given the following values:


n ( F ) = 60
n ( V ) = 48
n ( C ) = 40
n ( F o b re ak \tcap V ) = 32
n ( F o b re ak \tcap C ) = 22
n ( V o b re ak \tcap C ) = 20
n ( F o b re ak \tcap V o b re ak \tcap C ) = 5


Apply the formula and calculate the result Now, we substitute these values into the Principle of Inclusion-Exclusion formula:

n ( F o b re ak \tcup V o b re ak \tcup C ) = 60 + 48 + 40 − 32 − 22 − 20 + 5
n ( F o b re ak \tcup V o b re ak \tcup C ) = 148 − 74 + 5
n ( F o b re ak \tcup V o b re ak \tcup C ) = 74 + 5
n ( F o b re ak \tcup V o b re ak \tcup C ) = 79

State the final answer Therefore, the number of students who like at least one game is 79.

Examples
In a school, students can choose to participate in different clubs such as Math club, Science club, and Art club. Using the Principle of Inclusion-Exclusion, we can determine the total number of students involved in at least one of these clubs by considering the number of students in each club, the number in each pair of clubs, and the number in all three clubs. This helps the school administration understand the overall participation rate in extracurricular activities.
For example, if:

50 students are in Math club,
40 students are in Science club,
30 students are in Art club,
20 students are in both Math and Science clubs,
15 students are in both Math and Art clubs,
10 students are in both Science and Art clubs,
5 students are in all three clubs,

then the total number of students in at least one club is: 50 + 40 + 30 − 20 − 15 − 10 + 5 = 80 .

Answered by GinnyAnswer | 2025-07-03

Using the Principle of Inclusion-Exclusion, we calculated that the number of students who like at least one game is 79. We applied the formula to account for the overlaps between the groups of students interested in different sports. Overall, this demonstrates how to find the total in situations where individuals may belong to more than one group.
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Answered by Anonymous | 2025-07-04