Complete question is;
In a class of 40 students, 17 have ridden an airplane, 28 have ridden boat, 10 have ridden a train, 12 have ridden an airplane and a boat, 3 have ridden a train only, and 4 have ridden airplane only. Some students in the class have not ridden any of the three modes of transportation and an equal number have taken all three. How many students have used all three modes of transportation?
Answer:
4 students
Step-by-step explanation:
Let the number of students who used airplane be A
Let the number of students who used boat be B
Let the number of students who used train be C
Now, we are told that 17 rode plane.
Thus; A = 17
28 rode boat; B = 28
10 rode train; C = 10
12 rode airplane and boat; A ∩ B = 12
4 rode plane only; A' = 4
3 rode boat only; C' = 3
Total number of students; T = 40
Now, total number of students is represented by;
T = A - B - C - (A ∩ B) - (B ∩ C) - (C ∩ A) + (A ∩ B ∩ C)
We don't have (B ∩ C) and (C ∩ A).
Now, the can be derived from the expression of C' which is;
C' = C - (B ∩ C) - (C ∩ A)
C' = C - [(B ∩ C) + (C ∩ A)]
We are given C' = 3 and C = 10
Thus;
3 = 10 - [(B ∩ C) + (C ∩ A)]
10 - 3 = [(B ∩ C) + (C ∩ A)]
7 = [(B ∩ C) + (C ∩ A)]
Rearranging the total number of students equation, we now have;
T = A - B - C - (A ∩ B) - [(B ∩ C) + (C ∩ A)] + (A ∩ B ∩ C)
Where;
(A ∩ B ∩ C) is the number of students that used all three modes of transportation.
Thus, plugging in the relevant values;
40 = 17 + 28 + 10 - 12 - 7 + (A ∩ B ∩ C)
40 = 36 + (A ∩ B ∩ C)
(A ∩ B ∩ C) = 40 - 36
(A ∩ B ∩ C) = 4
