Given the problem, let's break it down and calculate the areas to find which sum of the areas of plots $P$, $Q$, and $R$ cannot occur.

Step 1: Calculate the areas of the square plots $A$, $B$, and $C$

• The area of plot $A$ is $a^2$ square meters.
• The area of plot $B$ is $b^2$ square meters.
• The area of plot $C$ is $c^2$ square meters.

It is given that the sum of these areas is:

$a^2 + b^2 + c^2 = 600 \text{ square meters}$

Step 2: Calculate the areas of the rectangular plots $P$, $Q$, and $R$

• The area of plot $P$ is $b \times a = ab$ square meters.
• The area of plot $Q$ is $c \times a = ac$ square meters.
• The area of plot $R$ is $c \times b = bc$ square meters.

The sum of the areas of $P$, $Q$, and $R$ is:

$\text{Sum of areas} = ab + ac + bc$

Step 3: Analyze possible values of $ab + ac + bc$

Given that $a^2 + b^2 + c^2 = 600$, and knowing that $a < b < c$, we can consider possible values for $a$, $b$, and $c$ that satisfy this equation.

1. Case 1: Assume $a = 10$, $b = 20$, and $c = 30$.

   • $a^2 + b^2 + c^2 = 10^2 + 20^2 + 30^2 = 100 + 400 + 900 = 1400$ (which is not possible as the sum exceeds 600).

2. Case 2: Assume $a = 5$, $b = 10$, and $c = 15$.

   • $a^2 + b^2 + c^2 = 5^2 + 10^2 + 15^2 = 25 + 100 + 225 = 350$ (which is less than 600, so this also does not satisfy).

Given the constraint $a^2 + b^2 + c^2 = 600$, let's calculate for typical values of $a$, $b$, and $c$.

Case 3: Suppose $a = 10$, $b = 15$, $c = 20$.

• $a^2 + b^2 + c^2 = 10^2 + 15^2 + 20^2 = 100 + 225 + 400 = 725$ (still too large).

So, based on typical integer values, let's check for other likely combinations.

Case 4: Assume $a = 10$, $b = 15$, $c = 17$.

• $a^2 + b^2 + c^2 = 10^2 + 15^2 + 17^2 = 100 + 225 + 289 = 614$ (this does not satisfy the equation as the total exceeds 600).

Conclusion:

Given that the possible values of $a$, $b$, and $c$ must satisfy $a^2 + b^2 + c^2 = 600$, and calculating the areas $ab + ac + bc$ for these values shows that the sum can vary. We need to check the possible sums that would violate this equation.

By checking specific possible values, the correct sum cannot be deduced from the given options in a general case, so careful checking of individual combinations is needed to identify specific scenarios where the sum $ab + ac + bc$ does not match the area requirement. Without specific values, we cannot pinpoint an impossible value directly.

For specific calculation with feasible values:

$\text{Sum of } ab + ac + bc = ?$

One would need to provide further constraints to identify a unique incorrect sum.

Would you like further details on these calculations or help in adjusting the values to find more accurate scenarios? Here are some related questions:

1. What are the exact values of $a$, $b$, and $c$ that satisfy $a^2 + b^2 + c^2 = 600$?
2. How do you systematically check all possible sums for such problems?
3. Can you derive general formulas for checking which sum is impossible?
4. How do constraints on $a < b < c$ affect the calculations?
5. How to derive the possible range of values for $ab + ac + bc$ given constraints?

Tip: For problems like these, try considering specific integer cases and checking them systematically against the given constraints.