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You're on a roll — Question 9: Let V represent volleyball, B represent basketball, and T represent table-tennis. Total number of females = 70. Each female plays at least one game, so n(V B T) = 70. Given information: n(V B T) = 20 (play all three games) n(B only) = 10 n(T only) = 6 n(V only) = 4 Let x be the number of females who play exactly two games for each pair. So, n(V B only) = x, n(V T only) = x, and n(B T only) = x. Step 1: Find the value of x. The sum of all disjoint regions in the Venn diagram must equal the total number of females. n(V B T) = n(V only) + n(B only) + n(T only) + n(V B only) + n(V T only) + n(B T only) + n(V B T) 70 = 4 + 10 + 6 + x + x + x + 20 70 = 40 + 3x 3x = 70 - 40 3x = 30 x = 10 So, n(V B only) = 10, n(V T only) = 10, and n(B T only) = 10. a) Illustrate this information on a Venn diagram. A Venn diagram would show three overlapping circles labeled V, B, and T. The numbers in each region would be: V only: 4 B only: 10 T only: 6 V B only: 10 V T only: 10 B T only: 10 V B T: 20 b) Find the number of females who play: i) volleyball; This includes those who play volleyball only, volleyball and basketball only, volleyball and table-tennis only, and all three. n(V) = n(V only) + n(V B only) + n(V T only) + n(V B T) n(V) = 4 + 10 + 10 + 20 n(V) = 44 The number of females who play volleyball is 44. ii) basketball; This includes those who play basketball only, volleyball and basketball only, basketball and table-tennis only, and all three. n(B) = n(B only) + n(V B only) + n(B T only) + n(V B T) n(B) = 10 + 10 + 10 + 20 n(B) = 50 The number of females who play basketball is 50. iii) table-tennis. This includes those who play table-tennis only, volleyball and table-tennis only, basketball and table-tennis only, and all three. n(T) = n(T only) + n(V T only) + n(B T only) + n(V B T) n(T) = 6 + 10 + 10 + 20 n(T) = 46 The number of females who play table-tennis is 46. c) how many females play exactly two games? This is the sum of the numbers in the regions where exactly two games are played. n(exactly two games) = n(V B only) + n(V T only) + n(B T only) n(exactly two games) = x + x + x = 3x Since x=10: n(exactly two games) = 3 × 10 = 30 The number of females who play exactly two games is 30. What's next?