If `n(xi)=40`, `n(A)=25` and `n(B)=20`, then find the lest value of `n(AnnB)`.

Answer

To find the least value of \( n(A \cap B) \), we can use the principle of inclusion-exclusion for sets. Let's break down the solution step by step.

### Step 1: Write down the given information
We have:
- \( n(U) = 40 \) (the number of elements in the universal set)
- \( n(A) = 25 \) (the number of elements in set A)
- \( n(B) = 20 \) (the number of elements in set B)

### Step 2: Use the formula for the union of two sets
The formula for the number of elements in the union of two sets is:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula gives us:
\[ n(A \cup B) = 25 + 20 - n(A \cap B) \]
This simplifies to:
\[ n(A \cup B) = 45 - n(A \cap B) \]

### Step 4: Set up the inequality based on the universal set
Since \( n(A \cup B) \) cannot exceed \( n(U) \), we have:
\[ 45 - n(A \cap B) \leq 40 \]

### Step 5: Rearrange the inequality
Rearranging the inequality gives:
\[ 45 - 40 \leq n(A \cap B) \]
This simplifies to:
\[ 5 \leq n(A \cap B) \]

### Step 6: Conclusion
Thus, the least value of \( n(A \cap B) \) is:
\[ n(A \cap B) \geq 5 \]
Therefore, the least value of \( n(A \cap B) \) is \( 5 \).

### Final Answer
The least value of \( n(A \cap B) \) is \( 5 \).

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