If odds against solving a question by three students are 2:1, 5:2, and 5:3, respectively, then probability that the question is solved only by one student is `31//56` b. `24//56` c. `25//56` d. none of these

Options

A. `31//56`
B. `24//56`
C. `23//56`
D. None of these

Explanation

To solve the problem, we first need to convert the odds against solving the question into probabilities for each student. The odds against solving a question by three students A, B, and C are given as:

- For student A: 2:1
- For student B: 5:2
- For student C: 5:3

### Step 1: Convert Odds to Probabilities
The odds against an event can be converted to probability using the formula:
\[ P(A) = \frac{\text{Odds in favor}}{\text{Total Odds}} \]

1. **For Student A:**
- Odds against = 2:1
- Odds in favor = 1 (solving) : 2 (not solving)
- Total = 1 + 2 = 3
- Probability \( P(A) = \frac{1}{3} \)

2. **For Student B:**
- Odds against = 5:2
- Odds in favor = 2 (solving) : 5 (not solving)
- Total = 2 + 5 = 7
- Probability \( P(B) = \frac{2}{7} \)

3. **For Student C:**
- Odds against = 5:3
- Odds in favor = 3 (solving) : 5 (not solving)
- Total = 3 + 5 = 8
- Probability \( P(C) = \frac{3}{8} \)

### Step 2: Calculate the Probabilities of Not Solving
Now, we calculate the probabilities of each student not solving the question:

- \( P(A') = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \)
- \( P(B') = 1 - P(B) = 1 - \frac{2}{7} = \frac{5}{7} \)
- \( P(C') = 1 - P(C) = 1 - \frac{3}{8} = \frac{5}{8} \)

### Step 3: Calculate the Probability that Only One Student Solves the Question
The probability that the question is solved by only one student can be expressed as:

\[ P(\text{only A solves}) + P(\text{only B solves}) + P(\text{only C solves}) \]

1. **Only A solves:**
\[ P(A) \cdot P(B') \cdot P(C') = \frac{1}{3} \cdot \frac{5}{7} \cdot \frac{5}{8} \]

2. **Only B solves:**
\[ P(A') \cdot P(B) \cdot P(C') = \frac{2}{3} \cdot \frac{2}{7} \cdot \frac{5}{8} \]

3. **Only C solves:**
\[ P(A') \cdot P(B') \cdot P(C) = \frac{2}{3} \cdot \frac{5}{7} \cdot \frac{3}{8} \]

### Step 4: Calculate Each Probability
Now we calculate each of these probabilities:

1. **Only A solves:**
\[ P(A \text{ only}) = \frac{1}{3} \cdot \frac{5}{7} \cdot \frac{5}{8} = \frac{25}{168} \]

2. **Only B solves:**
\[ P(B \text{ only}) = \frac{2}{3} \cdot \frac{2}{7} \cdot \frac{5}{8} = \frac{20}{168} \]

3. **Only C solves:**
\[ P(C \text{ only}) = \frac{2}{3} \cdot \frac{5}{7} \cdot \frac{3}{8} = \frac{30}{168} \]

### Step 5: Sum the Probabilities
Now we sum these probabilities to find the total probability that only one student solves the question:

\[ P(\text{only one solves}) = \frac{25}{168} + \frac{20}{168} + \frac{30}{168} = \frac{75}{168} \]

### Step 6: Simplify the Probability
Now we simplify \( \frac{75}{168} \):

\[ \frac{75}{168} = \frac{25}{56} \]

### Final Answer
Thus, the probability that the question is solved only by one student is \( \frac{25}{56} \).