A question is given to theree students A,B and C whose chances of solving it are `(1)/(2),(1)/(3)` and `(1)/(4)` respectively. What is the probaility that the question will be solved ?

Options

A. `(1)/(24)`
B. `(1)/(4)`
C. `(3)/(4)`
D. `(23)/(24)`

Explanation

To solve the problem, we need to find the probability that at least one of the students A, B, or C solves the question. We can use the complementary probability approach, which involves calculating the probability that none of the students solve the question and then subtracting that from 1.

### Step-by-Step Solution:

1. **Identify the probabilities of each student solving the question:**
- Probability that A solves the question, \( P(A) = \frac{1}{2} \)
- Probability that B solves the question, \( P(B) = \frac{1}{3} \)
- Probability that C solves the question, \( P(C) = \frac{1}{4} \)

2. **Calculate the probabilities of each student not solving the question:**
- Probability that A does not solve the question, \( P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \)
- Probability that B does not solve the question, \( P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \)
- Probability that C does not solve the question, \( P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4} \)

3. **Calculate the probability that none of the students solve the question:**
- Since the events are independent, the probability that none of them solve the question is given by:
\[ P(A' \cap B' \cap C') = P(A') \times P(B') \times P(C') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \]

4. **Perform the multiplication:**
- Calculate:
\[ P(A' \cap B' \cap C') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{1 \cdot 2 \cdot 3}{2 \cdot 3 \cdot 4} = \frac{6}{24} = \frac{1}{4} \]

5. **Calculate the probability that at least one student solves the question:**
- The probability that at least one student solves the question is:
\[ P(\text{at least one solves}) = 1 - P(A' \cap B' \cap C') = 1 - \frac{1}{4} = \frac{3}{4} \]

6. **Final Result:**
- The probability that the question will be solved is \( \frac{3}{4} \).