Question

Let us look at a class of students who play at least one of three sports - basketball, soccer and

hockey. Let the number of students playing basketball be 50, soccer be 60, and let hockey be
played by 40 students. Let the number of students who play both soccer and hockey be 12, the
number of students who play both basketball and soccer be 30. Let there be 20 students who
play basketball and hockey. Additionally, let us assume there are 10 very athletic students who
play all three sports. How many students are there in the class?

Answer 1

BB = 50

SOC = 60

HOC = 40

sum = 150

SOC + HOC = 12

BB + SOC = 30

BB + HOC = 20

BB + SOC + HOC = 10

first, these 10 we need to deduct twice from the total, as they were counted 3 times :

150 - 2×10 = 130

then the 12 of SOC + HOC minus the 10 BB + SOC + HOC = 2 were double counted and need to be removed once :

130 - 2 = 128

then the 30 of BB + SOC minus the 10 BB + SOC + HOC = 20 were double counted and need to be removed once :

128 - 20 = 108

then the 20 of BB + HOC minus the 10 BB + SOC + HOC = 10 were double counted and need to be removed once :

108 - 10 = 98

so, there were 98 students in the class.

FYI

so, there were

50 - (20 + 10) - 10 = 10 students playing only basketball.

60 - (2 + 20) - 10 = 28 students playing only soccer.

40 - (2 + 10) - 10 = 18 students playing only hockey.

in sum

10+28+18+2+20+10+10 = 98