A group of 70 females participates in volleyball, basketball, or table-tennis. If 20 play all three games, 10 play basketball only, 6 play table-tennis only, 4 play volleyball only, and the number of

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 $n(V \cup B \cup T) = 70$.

Given information:

• $n(V \cap B \cap T) = 20$ (play all three games)
• $n(B \text{ only}) = 10$
• $n(T \text{ only}) = 6$
• $n(V \text{ only}) = 4$
• Let $x$ be the number of females who play exactly two games for each pair. So, $n(V \cap B \text{ only}) = x$, $n(V \cap T \text{ only}) = x$, and $n(B \cap T \text{ only}) = x$.

Step 1: Find the value of $x$.
The sum of all disjoint regions in the Venn diagram must equal the total number of females.

$$n(V \cup B \cup T) = n(V \text{ only}) + n(B \text{ only}) + n(T \text{ only}) + n(V \cap B \text{ only}) + n(V \cap T \text{ only}) + n(B \cap T \text{ only}) + n(V \cap B \cap T)$$ $$70 = 4 + 10 + 6 + x + x + x + 20$$ $$70 = 40 + 3x$$ $$3x = 70 - 40$$ $$3x = 30$$ $$x = 10$$

So, $n(V \cap B \text{ only}) = 10$, $n(V \cap T \text{ only}) = 10$, and $n(B \cap T \text{ only}) = 10$.

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:

• $V \text{ only}: 4$
• $B \text{ only}: 10$
• $T \text{ only}: 6$
• $V \cap B \text{ only}: 10$
• $V \cap T \text{ only}: 10$
• $B \cap T \text{ only}: 10$
• $V \cap B \cap T: 20$

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.

$$n(V) = n(V \text{ only}) + n(V \cap B \text{ only}) + n(V \cap T \text{ only}) + n(V \cap B \cap T)$$ $$n(V) = 4 + 10 + 10 + 20$$ $$n(V) = 44$$

The number of females who play volleyball is $\boxed{44}$.

ii) basketball;
This includes those who play basketball only, volleyball and basketball only, basketball and table-tennis only, and all three.

$$n(B) = n(B \text{ only}) + n(V \cap B \text{ only}) + n(B \cap T \text{ only}) + n(V \cap B \cap T)$$ $$n(B) = 10 + 10 + 10 + 20$$ $$n(B) = 50$$

The number of females who play basketball is $\boxed{50}$.

iii) table-tennis.
This includes those who play table-tennis only, volleyball and table-tennis only, basketball and table-tennis only, and all three.

$$n(T) = n(T \text{ only}) + n(V \cap T \text{ only}) + n(B \cap T \text{ only}) + n(V \cap B \cap T)$$ $$n(T) = 6 + 10 + 10 + 20$$ $$n(T) = 46$$

The number of females who play table-tennis is $\boxed{46}$.

c) how many females play exactly two games?
This is the sum of the numbers in the regions where exactly two games are played.

$$n(\text{exactly two games}) = n(V \cap B \text{ only}) + n(V \cap T \text{ only}) + n(B \cap T \text{ only})$$ $$n(\text{exactly two games}) = x + x + x = 3x$$

Since $x=10$:

$$n(\text{exactly two games}) = 3 \times 10 = 30$$

The number of females who play exactly two games is $\boxed{30}$.