If A and B are two sets such that `n(A nn barB)= 9`, `n( barAnnB)= 10` and `n(AuuB)=24`, then `n(AxxB)=`

Options

A. 105
B. 210
C. 70
D. none of these

Answer

To solve the problem step by step, we will use the information given about the sets A and B and their intersections and unions.

### Step 1: Understand the Given Information
We are given:
- \( n(A \cap \bar{B}) = 9 \) (elements in A but not in B)
- \( n(\bar{A} \cap B) = 10 \) (elements in B but not in A)
- \( n(A \cup B) = 24 \) (elements in either A or B or both)

### Step 2: Use the Formula for Union of Sets
We know that:
\[ n(A \cup B) = n(A \cap \bar{B}) + n(\bar{A} \cap B) + n(A \cap B) \]
Substituting the known values into the equation:
\[ 24 = 9 + 10 + n(A \cap B) \]

### Step 3: Solve for \( n(A \cap B) \)
Now, we can simplify the equation:
\[ 24 = 19 + n(A \cap B) \]
Subtracting 19 from both sides gives:
\[ n(A \cap B) = 24 - 19 = 5 \]

### Step 4: Calculate the Number of Elements in Set A
To find the total number of elements in set A, we add the elements in \( A \cap \bar{B} \) and \( A \cap B \):
\[ n(A) = n(A \cap \bar{B}) + n(A \cap B) = 9 + 5 = 14 \]

### Step 5: Calculate the Number of Elements in Set B
Similarly, to find the total number of elements in set B, we add the elements in \( \bar{A} \cap B \) and \( A \cap B \):
\[ n(B) = n(\bar{A} \cap B) + n(A \cap B) = 10 + 5 = 15 \]

### Step 6: Calculate the Number of Elements in the Cartesian Product \( A \times B \)
The number of elements in the Cartesian product \( A \times B \) is given by:
\[ n(A \times B) = n(A) \times n(B) = 14 \times 15 \]
Calculating this gives:
\[ n(A \times B) = 210 \]

### Final Answer
Thus, the number of elements in \( A \times B \) is \( 210 \).

---