In a regional sports academy, a group of 70 trainees were surveyed about their participati… — State PSC Exam

In a regional sports academy, a group of $70$ trainees were surveyed about their participation in two sports: Cricket and Football. $45$ trainees play Cricket, $30$ trainees play Football, and $10$ trainees play neither sport. How many trainees play both Cricket and Football?

$15$ Answer
B
$5$
C
$25$
D
$10$

Answer

15 trainees play both Cricket and Football.

The total number of trainees participating in at least one sport is found by subtracting the

10

non-participants from the total

70

, giving

60

. Summing the individual sport counts gives

45 + 30 = 75

. The overlap represents those counted twice, which is

75 - 60 = 15

Step-by-Step Solution

Calculate the total number of trainees who play at least one sport.

Trainees playing at least one sport =

70 - 10 = 60

Subtract trainees playing neither sport from the total group size.

Apply the Principle of Inclusion-Exclusion for two sets.

N(\text{Cricket} \cup \text{Football}) = N(\text{Cricket}) + N(\text{Football}) - N(\text{Cricket} \cap \text{Football})

The sum of individual sets double-counts the intersection.

Substitute the known values into the equation and solve for the intersection.

60 = 45 + 30 - N(\text{Cricket} \cap \text{Football}) \implies N(\text{Cricket} \cap \text{Football}) = 75 - 60 = 15

Algebraic simplification yields the number of trainees playing both sports.

Key Concept

Two-Set Inclusion-Exclusion Principle

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