A problem in mathematics is given to three students and their respective probabilities of solving the problem are `(1)/(2)` ,`(1)/(3)`and `(1)/(4)`. The probability that the problem is solved, is

Options

A. `(3)/(4)`
B. `(1)/(2)`
C. `(2)/(3)`
D. `(1)/(3)`

Answer

To solve the problem, we need to find the probability that at least one of the three students solves the problem. We will use the complementary probability approach, which involves calculating the probability that none of the students solve the problem and then subtracting this value from 1.

### Step-by-Step Solution:

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

2. **Calculate the probabilities that each student does not solve the problem:**
- The probability that student A does not solve the problem is:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \]
- The probability that student B does not solve the problem is:
\[ P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \]
- The probability that student C does not solve the problem is:
\[ P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4} \]

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

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

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

### Final Answer:
The probability that the problem is solved is \( \frac{3}{4} \).

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