n(A)=1400,n(B)=1500and n(U)=2855 then `n(A cap B)`=

Options

A. 45
B. 36
C. 55
D. None of these

Answer

To solve the problem step by step, we will use the principle of set theory involving the union and intersection of sets.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- \( n(A) = 1400 \)
- \( n(B) = 1500 \)
- \( n(U) = 2855 \) (This represents \( n(A \cup B) \))

2. **Use the Formula for Union of Two Sets:**
The formula for the union of two sets is given by:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Here, \( n(A \cup B) \) is the total number of elements in either set A or set B or both.

3. **Substitute the Known Values into the Formula:**
We know \( n(A \cup B) = 2855 \), \( n(A) = 1400 \), and \( n(B) = 1500 \). Plugging these values into the formula gives:
\[ 2855 = 1400 + 1500 - n(A \cap B) \]

4. **Simplify the Equation:**
Combine the values on the right side:
\[ 2855 = 2900 - n(A \cap B) \]

5. **Rearrange the Equation to Solve for \( n(A \cap B) \):**
To isolate \( n(A \cap B) \), we can rearrange the equation:
\[ n(A \cap B) = 2900 - 2855 \]

6. **Calculate \( n(A \cap B) \):**
Now, perform the subtraction:
\[ n(A \cap B) = 45 \]

### Final Answer:
Thus, the number of elements in the intersection of sets A and B is:
\[ n(A \cap B) = 45 \]