Question

In a club of 80 members, 10 members play none of the games Tennis, Badminton and Cricket. 30 members play exactly one of these three games and 30 members play exactly two of these games. 45 members play at least one of the games among Tennis, and Badminton, whereas 18 members play both Tennis and Badminton. Determine the number of Cricket playing members.

Answer 1

See the Venn diagram enclosed for the sets.
We can use the set notation, with unions and intersections and formulas. The following is equivalent to that.

Total N = 80                                  e = 10
x + y + z = 30                               a + b + c = 30
x + y + a + b + c + d = 45             c + d = 18
=>  x + y + d = 45 - 30 =15

x+y+ z+ a+ b+ c + d = N- e = 80 - 10 = 70
=>  (x+y+d) + z + (a+b+c) = 70
=> 15 + z + 30 = 70
=>  z = 25
=> x+y = 30 - 25 = 5

Also,   x + y + z + a + b + c+ d = 70
           30 + 30 + d = 70    =>  d = 10
          =>  c = 18 - d = 18 - 10 = 8

Now ,  a+ b = 30 - c = 22 

Number of players playing cricket = z + d + a+b = 25 + 10 + 22
    = 57