If the difference between the number of subsets of two sets A and B is 120, then `n(A xx B)` is equal to

Options

A. 21
B. 25
C. 18
D. 24

Answer

To solve the problem, we need to find the value of \( n(A \times B) \) given that the difference between the number of subsets of two sets \( A \) and \( B \) is 120. Let's denote the number of elements in set \( A \) as \( P \) and in set \( B \) as \( Q \).

### Step 1: Understand the Number of Subsets
The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore:
- The number of subsets of set \( A \) is \( 2^P \).
- The number of subsets of set \( B \) is \( 2^Q \).

### Step 2: Set Up the Equation
According to the problem, the difference between the number of subsets of sets \( A \) and \( B \) is 120. We can express this as:
\[ 2^P - 2^Q = 120 \]

### Step 3: Assume \( P > Q \)
For simplicity, let's assume \( P > Q \). We can factor out \( 2^Q \) from the left-hand side:
\[ 2^Q (2^{P-Q} - 1) = 120 \]

### Step 4: Factorize 120
Next, we can factor 120 to find possible values for \( 2^Q \):
\[ 120 = 2^3 \times 15 \]
This gives us \( 2^Q = 8 \) (which means \( Q = 3 \)) and \( 2^{P-Q} - 1 = 15 \).

### Step 5: Solve for \( P \)
From \( 2^{P-Q} - 1 = 15 \):
\[ 2^{P-Q} = 16 \implies P - Q = 4 \]
Since we have \( Q = 3 \):
\[ P - 3 = 4 \implies P = 7 \]

### Step 6: Calculate \( n(A \times B) \)
Now that we have \( P = 7 \) and \( Q = 3 \), we can find \( n(A \times B) \):
\[ n(A \times B) = n(A) \cdot n(B) = P \cdot Q = 7 \cdot 3 = 21 \]

### Final Answer
Thus, the value of \( n(A \times B) \) is \( \boxed{21} \).

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