Question

In a survey of students about favorite​ sports, the results include 18 who like​ tennis, 25 who like​ football, 10 who like tennis and​ football, 13 who like tennis and​ baseball, 20 who like football and​ baseball, 8 who like all three​ sports, and 9 who like none of the sports. How many students like only tennis and​ football? How many students like only tennis and​ baseball? How many students like only baseball and​ football?

Answer 1

Number of students who like only tennis and football = 2

Number of students who like only tennis and baseball = 5

Number of students who like only baseball and football = 12

Solution:

Let students who likes tennis be represented by set T.

Let students who likes football be represented by set F.

Let students who likes baseball be represented by set B.

Given that

Number of students who likes tennis that is n(T) = 18

Number of students who likes football that is n(F) = 25

Number of students who likes tennis and football = n( T ∩ F ) = 10

Number of students who likes tennis and baseball = n( T ∩ B ) = 13

Number of students who likes football and baseball = n( F ∩ B ) = 20

Number of students who likes all three sports = n(T ∩ F ∩ B ) = 8

Number of students who likes none of the sports = 9.

Determining number of students who like only tennis and football:

number of students who like only tennis and football = Number of students who likes tennis and football - Number of students who likes all three sports

so number of students who like only tennis and football = n( T ∩ F ) - n(T ∩ F ∩ B )

= 10 – 8 = 2

Determining number of students who like only tennis and baseball:

number of students who like only tennis and baseball = Number of students who likes tennis and baseball - Number of students who likes all three sports

so number of students who like only tennis and baseball = n( T ∩ B ) - n(T ∩ F ∩ B )

= 13 – 8 = 5

Determining number of students who like only baseball and football:

number of students who like only baseball and football = Number of students who likes baseball and football - Number of students who likes all three sports

so number of students who like only baseball and football = n( B∩ F ) - n(T ∩ F ∩ B)

= 20 – 8 = 12

Summarizing the result :

number of students who like only tennis and football = 2

number of students who like only tennis and baseball = 5

number of students who like only baseball and football = 12.