Suppose A is a set containing n elements. Two subsets P and Q of A are chosen at random. If the probability that P is a subset of Q is 243/1024, then n is equal to

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem, we need to find the value of \( n \) given that the probability that subset \( P \) is a subset of subset \( Q \) is \( \frac{243}{1024} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a set \( A \) with \( n \) elements. We randomly choose two subsets \( P \) and \( Q \) from \( A \). We need to find the probability that \( P \subseteq Q \).

2. **Total Number of Subsets**:
The total number of subsets of a set with \( n \) elements is \( 2^n \). Therefore, the total number of ways to choose subsets \( P \) and \( Q \) is:
\[ 2^n \times 2^n = 4^n \]

3. **Favorable Outcomes**:
For \( P \) to be a subset of \( Q \), we consider how many elements can be chosen for \( Q \) and how many can be chosen for \( P \) based on the elements in \( Q \):
- If \( Q \) has \( k \) elements, \( P \) can have any subset of those \( k \) elements, including the empty set. The number of subsets of \( k \) elements is \( 2^k \).
- The number of ways to choose \( k \) elements from \( n \) is \( \binom{n}{k} \).

Therefore, the total number of favorable outcomes where \( P \subseteq Q \) is:
\[ \sum_{k=0}^{n} \binom{n}{k} 2^k \]
This expression represents choosing \( k \) elements for \( Q \) and then choosing any subset of those \( k \) elements for \( P \).

4. **Using the Binomial Theorem**:
The sum can be simplified using the Binomial Theorem:
\[ \sum_{k=0}^{n} \binom{n}{k} 2^k = (1 + 2)^n = 3^n \]

5. **Calculating the Probability**:
The probability that \( P \subseteq Q \) is given by the ratio of favorable outcomes to total outcomes:
\[ P(P \subseteq Q) = \frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n \]

6. **Setting Up the Equation**:
We know from the problem statement that:
\[ \left(\frac{3}{4}\right)^n = \frac{243}{1024} \]

7. **Expressing the Right Side in Powers**:
We can express \( 243 \) and \( 1024 \) as powers:
\[ 243 = 3^5 \quad \text{and} \quad 1024 = 4^5 \]
Thus, we can rewrite the equation as:
\[ \left(\frac{3}{4}\right)^n = \left(\frac{3}{4}\right)^5 \]

8. **Equating the Exponents**:
Since the bases are the same, we can equate the exponents:
\[ n = 5 \]

### Final Answer:
Thus, the value of \( n \) is \( 5 \).