Let `A={a,b,c} , B={5,7},` and set C be a set containing n elements such that `B cap C = phi`. If `A xx (B cup C)` has 33 elements, then n is equal to

Options

A. 7
B. 8
C. 9
D. 11

Answer

To solve the problem step by step, we will follow the logical flow of the given information.

### Step 1: Understand the Sets
We have three sets:
- \( A = \{a, b, c\} \) which contains 3 elements.
- \( B = \{5, 7\} \) which contains 2 elements.
- \( C \) is a set that contains \( n \) elements, and it must not share any elements with set \( B \) (i.e., \( B \cap C = \emptyset \)).

### Step 2: Calculate the Number of Elements in \( B \cup C \)
Since \( B \) and \( C \) have no common elements, the number of elements in the union \( B \cup C \) can be calculated as:
\[ |B \cup C| = |B| + |C| = 2 + n \]
where \( |B| = 2 \) (the number of elements in \( B \)) and \( |C| = n \) (the number of elements in \( C \)).

### Step 3: Calculate the Number of Elements in \( A \times (B \cup C) \)
The Cartesian product \( A \times (B \cup C) \) will have:
\[ |A \times (B \cup C)| = |A| \times |(B \cup C)| \]
where \( |A| = 3 \) (the number of elements in \( A \)). Therefore, we have:
\[ |A \times (B \cup C)| = 3 \times (2 + n) \]

### Step 4: Set the Equation
According to the problem, the total number of elements in \( A \times (B \cup C) \) is given as 33. Thus, we can set up the equation:
\[ 3 \times (2 + n) = 33 \]

### Step 5: Solve for \( n \)
To solve for \( n \), we first divide both sides of the equation by 3:
\[ 2 + n = \frac{33}{3} \]
\[ 2 + n = 11 \]
Now, we subtract 2 from both sides:
\[ n = 11 - 2 \]
\[ n = 9 \]

### Conclusion
The value of \( n \) is 9.