SOLUTION: QUESTION TIME!~ in a class of 40 students, 17 have ridden an airplane 28 have ridden a boat 10 have ridden a train 12 have ridden both airplane and a boat 3 have ridden a train

Question 627583: QUESTION TIME!~
in a class of 40 students, 17 have ridden an airplane 28 have ridden a boat 10 have ridden a train 12 have ridden both airplane and a boat 3 have ridden a train only 4 have ridden an airplane only
some students in the class have not ridden any of the modes of transportation. QUESTION: how many students have taken all three transportation?
Answer by psbhowmick(878) (Show Source): You can put this solution on YOUR website!
Let the no. of students who have ridden airplane be A, the no. of students who have ridden boat be B and that who have ridden train be C.

Further, let the no. of students who have ridden only airplane be A', the no. of students who have ridden only boat be B' and that who have ridden only train be C'.
If the total number of students be N then we have
N = A' + B' + C' + A∩B + B∩C + C∩A + A∩B∩C ________ (1)

Also, we know
A' = A - A∩B - C∩A ________ (2)
B' = B - A∩B - B∩C ________ (3)
C' = C - B∩C - C∩A ________ (4)

Substituting in the equation (1) we have
N = A + B + C - A∩B - B∩C - C∩A + A∩B∩C __________ (5)
Now let us list all the values that we have:
N = 40, A = 17, B = 28, C = 10, A∩B = 12, C' = 3, A' = 4

Substituting all these values into equation (5)
40 = 17 + 28 + 10 - 12 - (B∩C + C∩A) + A∩B∩C
40 = 17 + 28 + 10 - 12 - (C - C') + A∩B∩C [Using equation (4)]
40 = 17 + 28 + 10 - 12 - (10 - 3) + A∩B∩C
A∩B∩C = 4

Answer: The no. of students who have ridden all modes of transportation is 4.
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