In check up of 100 injured persons, injured in an accident in an industry, the number of injured persons by eye, hand and leg are respectively 20,50 and 70. The 44 injured persons are injured by eye and hand or hand and leg or eye and leg. Find the number of injured persons who are injured by all eye, hand and leg.

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion for sets.

### Step 1: Define the Sets
Let:
- \( E \) = Set of injured persons by eye
- \( H \) = Set of injured persons by hand
- \( L \) = Set of injured persons by leg

From the problem, we have:
- \( |E| = 20 \) (injured by eye)
- \( |H| = 50 \) (injured by hand)
- \( |L| = 70 \) (injured by leg)

### Step 2: Define the Intersections
Let:
- \( x \) = Number of injured persons who are injured by all three (eye, hand, and leg).
- The number of injured persons who are injured by eye and hand or hand and leg or eye and leg is given as 44.

Using the principle of inclusion-exclusion, we can express this as:
\[ |H \cap E| + |H \cap L| + |L \cap E| - 2x = 44 \]
where \( |H \cap E| \), \( |H \cap L| \), and \( |L \cap E| \) are the numbers of injured persons in the intersections of the respective sets.

### Step 3: Total Number of Injured Persons
The total number of injured persons is given as 100:
\[ |H \cup E \cup L| = |H| + |E| + |L| - |H \cap E| - |H \cap L| - |L \cap E| + x = 100 \]

### Step 4: Substitute Known Values
Substituting the known values into the equations:
1. From the total injured persons:
\[ 50 + 20 + 70 - (|H \cap E| + |H \cap L| + |L \cap E|) + x = 100 \]
Simplifying this gives:
\[ 140 - (|H \cap E| + |H \cap L| + |L \cap E|) + x = 100 \]
Rearranging gives:
\[ |H \cap E| + |H \cap L| + |L \cap E| - x = 40 \quad \text{(1)} \]

2. From the intersections:
\[ |H \cap E| + |H \cap L| + |L \cap E| - 2x = 44 \quad \text{(2)} \]

### Step 5: Solve the Equations
Now we have two equations:
1. \( |H \cap E| + |H \cap L| + |L \cap E| - x = 40 \)
2. \( |H \cap E| + |H \cap L| + |L \cap E| - 2x = 44 \)

Subtract equation (1) from equation (2):
\[ (|H \cap E| + |H \cap L| + |L \cap E| - 2x) - (|H \cap E| + |H \cap L| + |L \cap E| - x) = 44 - 40 \]
This simplifies to:
\[ -x = 4 \implies x = 4 \]

### Conclusion
The number of injured persons who are injured by all three (eye, hand, and leg) is \( x = 4 \).

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