In a battle, `70%` of the combatants lost one eye, `80%` an ear, `75%` an arm. `85%` a leg, and `x%` lost all the four organs. Then minimum value of x is

Options

A. 8
B. 12
C. 15
D. 10

Answer

To solve the problem, we need to find the minimum percentage \( x \) of combatants who lost all four organs: one eye, one ear, one arm, and one leg. We will use the principle of inclusion-exclusion.

### Step-by-Step Solution:

1. **Define the Percentages**:
- Let \( A \) be the set of combatants who lost one eye: \( |A| = 70\% \)
- Let \( B \) be the set of combatants who lost one ear: \( |B| = 80\% \)
- Let \( C \) be the set of combatants who lost one arm: \( |C| = 75\% \)
- Let \( D \) be the set of combatants who lost one leg: \( |D| = 85\% \)

2. **Use the Inclusion-Exclusion Principle**:
The principle states:
\[ |A \cup B \cup C \cup D| = |A| + |B| + |C| + |D| - |A \cap B| - |A \cap C| - |A \cap D| - |B \cap C| - |B \cap D| - |C \cap D| + |A \cap B \cap C| + |A \cap B \cap D| + |A \cap C \cap D| + |B \cap C \cap D| - |A \cap B \cap C \cap D| \]

3. **Calculate the Total Percentage**:
Since the total percentage of combatants cannot exceed 100%, we have:
\[ |A| + |B| + |C| + |D| - |A \cap B| - |A \cap C| - |A \cap D| - |B \cap C| - |B \cap D| - |C \cap D| + |A \cap B \cap C| + |A \cap B \cap D| + |A \cap C \cap D| + |B \cap C \cap D| - |A \cap B \cap C \cap D| \leq 100 \]

4. **Substituting Known Values**:
We know:
\[ |A| + |B| + |C| + |D| = 70 + 80 + 75 + 85 = 310 \]
Let \( x = |A \cap B \cap C \cap D| \). Then we can express the equation as:
\[ 310 - (|A \cap B| + |A \cap C| + |A \cap D| + |B \cap C| + |B \cap D| + |C \cap D|) + (|A \cap B \cap C| + |A \cap B \cap D| + |A \cap C \cap D| + |B \cap C \cap D|) - x \leq 100 \]

5. **Finding the Minimum Value of \( x \)**:
To find the minimum value of \( x \), we can assume the worst-case scenario where the overlaps are minimized. The maximum possible overlaps can be calculated as follows:
- The maximum overlap of any two sets cannot exceed the smaller of the two percentages.
- For example, \( |A \cap B| \) can be at most \( \min(70, 80) = 70\% \).

Continuing this process, we can find that the minimum value of \( x \) that satisfies the inequality is:
\[ x \geq 10\% \]

Thus, the minimum value of \( x \) is **10%**.