Agroup of 72 children completed a survey on what kind of sport they like. the choices were: chess, swimming, and football. everyone liked at least one sport except 7 kids, which doesn't like any of these three kind of sports. 12 children liked chess and football but not swimming, 16 children liked chess and swimming but not football, 8 children liked swimming and football but not chess, 10 children liked chess only, 40 children liked swimming, 32 student liked football. a) find the number of kids who liked swimming and football. b)what is the probability that a randomly-chosen child from this group likes either chess or swimming but not football? c)what is the probability that a randomly-chosen child from this group not likes active kinds of sport? the answer to a) is not 64 or 8.

The attached shows the Venn diagram for the current problem.
C = Chess
F = Football
S = Swimming
Total number of children = 72
Those who don't like any of the sports = 7

(a) Children who like swimming and football
Then,
40 children like swimming, then
S+16+x+8 = 40 => S+x = 16 --- (1)
32 children like football, then
F+8+x+12 = 32 => F+x = 12 --- (2)
But, the total of those who like sports is 65 (that is, 72 -7). Then
S+16+x+8+12+F+10 = 65 => S+F+x = 19 ---- (3)

From (1), S = 16 -x
Using (3), 16-x+F+x = 19 => 16+F = 19 => F = 3
Using (2), 3+x=12 => x = 12-3 = 9
And, S = 16-x = 16 -9 = 7

Therefore,
S (those who like swimming) = 7 children
F (those who like football) = 3 children

(b) Child chosen randomly likes Chess of  Swimming but not Football

P(C) = 10/72 = 5/36
P(S) = 3/72 = 1/24

P(C) or P(S) = 5/36 + 1/24 = 13/72

(c) Child chosen dos not like sport
P(No sport) = 7/72