In a singing competition a group of 10 pepole, participate, each person has exactly one friend and rest are strangers, the probability that both the singers of the duet are complete strangers is

Options

A. `1/15`
B. `1/45`
C. `8/9`
D. `22/45`

Answer

To solve the problem, we need to find the probability that both singers chosen for the duet are complete strangers in a group of 10 people, where each person has exactly one friend and the rest are strangers.

### Step-by-Step Solution:

1. **Understanding the Group Dynamics**:
- There are 10 people in total.
- Each person has exactly one friend, which means they are paired up. Thus, we can think of the group as consisting of 5 pairs of friends.

2. **Total Ways to Choose 2 People**:
- The total number of ways to choose 2 people from 10 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of people and \( r \) is the number of people to choose.
- Therefore, the total ways to choose 2 people from 10 is:
\[ \text{Total ways} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]

3. **Choosing Complete Strangers**:
- To ensure that both chosen individuals are complete strangers, we can select one person and then choose a second person who is not their friend.
- If we choose one person (10 options), their friend cannot be chosen, leaving us with 8 possible strangers to choose from.
- Thus, the number of favorable outcomes for choosing 2 strangers is:
\[ \text{Favorable outcomes} = 10 \times 8 = 80 \]

4. **Adjusting for Overcounting**:
- Since the order of selection does not matter (choosing person A and then person B is the same as choosing person B and then person A), we need to divide by 2 to correct for this overcounting:
\[ \text{Adjusted favorable outcomes} = \frac{80}{2} = 40 \]

5. **Calculating the Probability**:
- The probability \( P \) that both singers chosen are complete strangers is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Adjusted favorable outcomes}}{\text{Total ways}} = \frac{40}{45} = \frac{8}{9} \]

### Final Answer:
The probability that both singers of the duet are complete strangers is \( \frac{8}{9} \).