Question

In a class of 60 students, 23 play hockey, 15 play basketball,20 play cricket and 7 play hockey and basketball, 5 play cricket and basketball, 4 play hockey and cricket, 15 do not play any of the three games. Find

(i) How many play hockey, basketball and cricket

(ii) How many play hockey but not cricket

(iii) How many play hockey and cricket but not basketball

Answer 1

Let B,C Nd H be the sets of students that play basketball, cricket Nd hockey

n(H) = 23 , n(B) = 15 , n(c) = 20

n(H intersection B) =7 , n(c intersection B) = 5 , n(H intersection C) = 4

n(H union B Union C ) = 60-15= 45

• 1 Students that plays all games

n (H intersection B intersection C) = n(H union B Union) - n(H) - n(B) -n(C) +n( H intersection B) + n( B intersection C) + n( C intersection H)

= 45 - 23 - 15 - 20 +7 + 5 + 4 = 3

• 2 who plays hockey not crickret

n( H) - n(H intersection C) = 23-4 = 19

• Plays cricket Nd hockey but not basketball

n(H intersection C) - n( H intersection C intersection B) = 4-3=1

Answer 2

We know that the total number of students in the is class is 60 but out of these 60 students 15 students do not play any of the games so now only 60 - 15 = 45 students play the games

a+e+g+f=23 ( total hockey) (1)b+d+g+f=15 (total basketball ) (2)c+e+g+d=20 ( total cricket) (3)And as it is also given thate + g = 4( cricket and hockey)f + g = 7(hockey and basketball)d+ g = 5 (cricket and basketball)Now a+b+c+d+e+ f + g = total number of students that play any of the gamesa + b + c + d + e + f + g = 60 - 15 = 45studentsNow by putting the value of f + g = 7

16 and by putting the value of d + g = 5 in equation (3) we will get c + e = 15.And similarly by following the substitution method we will get b+d=8, b+f=10, c+d=16, and also we get b + c + d = 22, a +c+e= 30 and a + b + f=25 and from by solving all the above equations by putting the value of all the equations we will get b = 6a = 15C = 14

e = 1

e = 1d = 2

e = 1d = 2f = 4 and g = 3

e = 1d = 2f = 4 and g = 3And from all the above values we come to know that as e (i.e. students who play hockey and cricket only = 1 )

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