Problem 11

Explanation

Given:

• Total students $= 700$.
• Students choosing sports $= 350$.
• Students choosing arts $= 450$.

We want to find the value of $P$ (maximum intersection) and $Q$ (minimum intersection).

1. Finding $P$ (Maximum Intersection) The maximum number of students participating in both occurs if one set is a subset of the other (or as many members of the smaller set as possible are included in the larger set).

   $$P = \min(350, 450) = 350 \text{ students}$$

2. Finding $Q$ (Minimum Intersection) The minimum number of students participating in both occurs if the distribution of students is as spread out as possible so that the union of the two sets is maximized (equal to the total universe, which is $700$). Using the Inclusion-Exclusion Principle:

   $$\text{Intersection} = (\text{Sports} + \text{Arts}) - \text{Total}$$

   $$Q = (350 + 450) - 700$$

   $$Q = 800 - 700 = 100 \text{ students}$$

3. Finding the Difference Between $P$ and $Q$

   $$P - Q = 350 - 100 = 250 \text{ students}$$

Thus, the difference between $P$ and $Q$ is $250$ students.