A problem is given to three students whose chances of solving it are `1/4, 1/5 and 1/3` respectively. Find the probability that the problem is solved.

Answer

To solve the problem, we need to find the probability that at least one of the three students solves the problem. The probabilities of each student solving the problem are given as follows:

- Probability that student A solves the problem, \( P(A) = \frac{1}{4} \)
- Probability that student B solves the problem, \( P(B) = \frac{1}{5} \)
- Probability that student C solves the problem, \( P(C) = \frac{1}{3} \)

### Step 1: Find the probabilities that each student does not solve the problem.

The probability that student A does not solve the problem is:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4} \]

The probability that student B does not solve the problem is:
\[ P(B') = 1 - P(B) = 1 - \frac{1}{5} = \frac{4}{5} \]

The probability that student C does not solve the problem is:
\[ P(C') = 1 - P(C) = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 2: Find the probability that none of the students solve the problem.

To find the probability that none of the students solve the problem, we multiply the probabilities that each student does not solve it:
\[ P(A' \cap B' \cap C') = P(A') \times P(B') \times P(C') = \frac{3}{4} \times \frac{4}{5} \times \frac{2}{3} \]

### Step 3: Simplify the expression.

Calculating the multiplication:
\[ P(A' \cap B' \cap C') = \frac{3 \times 4 \times 2}{4 \times 5 \times 3} \]
The \(3\) in the numerator and denominator cancels out:
\[ = \frac{4 \times 2}{4 \times 5} = \frac{2}{5} \]

### Step 4: Find the probability that at least one student solves the problem.

The probability that at least one student solves the problem is given by:
\[ P(\text{at least one solves}) = 1 - P(A' \cap B' \cap C') \]
Substituting the value we found:
\[ P(\text{at least one solves}) = 1 - \frac{2}{5} = \frac{3}{5} \]

### Final Answer:
The probability that the problem is solved is:
\[ \frac{3}{5} \]