The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :

Options

A. `(65)/(2^7)`
B. `(65)/(2^8)`
C. `(135)/(2^9)`
D. `(35)/(2^7)`

Answer

To find the probability that two randomly selected subsets of the set \( S = \{1, 2, 3, 4, 5\} \) have exactly two elements in their intersection, we can follow these steps:

### Step 1: Determine the total number of subsets
Each element in the set \( S \) can either be included in a subset or not. Therefore, for a set with \( n \) elements, the total number of subsets is \( 2^n \). For our set \( S \) with 5 elements:

\[ \text{Total subsets} = 2^5 = 32 \]

### Step 2: Calculate the total number of ways to select two subsets
Since we are selecting two subsets from the total of 32 subsets, the total number of ways to select two subsets (A and B) is:

\[ \text{Total ways to select two subsets} = 32 \times 32 = 1024 \]

### Step 3: Determine the favorable conditions
We need to find the number of ways to select subsets A and B such that they have exactly 2 elements in common.

1. **Choose 2 elements for the intersection**: We can choose 2 elements from the 5 elements in the set. The number of ways to choose 2 elements from 5 is given by the combination formula \( \binom{n}{r} \):

\[ \binom{5}{2} = 10 \]

2. **Assign the remaining elements**: After choosing 2 elements for the intersection, we have 3 elements left. Each of these remaining elements can either:
- Be included in subset A only,
- Be included in subset B only,
- Be included in neither subset.

Thus, each of the 3 remaining elements has 3 choices. Therefore, the total number of ways to assign these 3 elements is:

\[ 3^3 = 27 \]

### Step 4: Calculate the total favorable outcomes
Now, we can calculate the total number of favorable outcomes where subsets A and B have exactly 2 elements in common:

\[ \text{Favorable outcomes} = \text{Ways to choose intersection} \times \text{Ways to assign remaining elements} = 10 \times 27 = 270 \]

### Step 5: Calculate the probability
The probability \( P \) that two randomly selected subsets have exactly two elements in their intersection is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P = \frac{\text{Favorable outcomes}}{\text{Total ways to select two subsets}} = \frac{270}{1024} \]

### Step 6: Simplify the probability
To simplify \( \frac{270}{1024} \):

1. Find the greatest common divisor (GCD) of 270 and 1024. The GCD is 2.
2. Divide both the numerator and the denominator by 2:

\[ P = \frac{135}{512} \]

Thus, the final answer is:

\[ \text{Probability} = \frac{135}{512} \]