In a survey of 90 high school students regarding their participation in sports clubs, 40 p… — JAMB UTME

In a survey of $90$ high school students regarding their participation in sports clubs, $40$ play Badminton ( $B$ ), $35$ play Volleyball ( $V$ ), and $42$ engage in Swimming ( $S$ ). It was found that $14$ play both Badminton and Volleyball, $12$ play both Volleyball and Swimming, and $15$ play both Badminton and Swimming. If $8$ students participate in none of these three sports, how many students participate in all three sports?

$6$ Answer
B
$14$
C
$10$
D
$18$

Answer

The number of students participating in all three sports is

6

First, determine the number of students in the union of the three sports sets by subtracting the

8

non-participating students from the universal set of

90

, giving

82

. Then apply the Principle of Inclusion-Exclusion:

82 = 40 + 35 + 42 - (14 + 12 + 15) + x

, where

x

is the number of students participating in all three sports. Simplifying gives

82 = 76 + x

, which yields

x = 6

Step-by-Step Solution

Calculate the cardinality of the union of the three sets.

n(B \cup V \cup S) = n(U) - n((B \cup V \cup S)') = 90 - 8 = 82

Subtracting the students who do not participate in any sport from the total universal set gives the number of students participating in at least one sport.

Set up the Principle of Inclusion-Exclusion formula for three sets.

n(B \cup V \cup S) = n(B) + n(V) + n(S) - [n(B \cap V) + n(V \cap S) + n(B \cap S)] + n(B \cap V \cap S)

This fundamental relation accounts for overlapping subsets in a three-set system.

Substitute the known numerical values into the formula and solve for

n(B \cap V \cap S)

82 = 40 + 35 + 42 - (14 + 12 + 15) + n(B \cap V \cap S) \implies 82 = 117 - 41 + n(B \cap V \cap S) \implies 82 = 76 + n(B \cap V \cap S) \implies n(B \cap V \cap S) = 82 - 76 = 6

Simplifying the arithmetic expression directly isolates the unknown intersection value.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic