Probability of Solving the Problem

To find the probability that the problem will be solved by at least one student, we can use the following formula:

P(at least one student solves the problem) = 1 - P(no student solves the problem)

So, we need to find the probability that no student solves the problem and subtract it from 1 to get the probability that at least one student solves the problem.

Probability that No Student Solves the Problem

The probability that no student solves the problem is the product of the probabilities that each student does not solve the problem. So, we can use the following formula:

P(no student solves the problem) = P(student 1 does not solve the problem) x P(student 2 does not solve the problem) x P(student 3 does not solve the problem)

We are given the individual probabilities of solving the problem for each student as follows:

- P(student 1 solves the problem) = 1/2
- P(student 2 solves the problem) = 1/3
- P(student 3 solves the problem) = 1/4

So, the probabilities that each student does not solve the problem are:

- P(student 1 does not solve the problem) = 1 - P(student 1 solves the problem) = 1 - 1/2 = 1/2
- P(student 2 does not solve the problem) = 1 - P(student 2 solves the problem) = 1 - 1/3 = 2/3
- P(student 3 does not solve the problem) = 1 - P(student 3 solves the problem) = 1 - 1/4 = 3/4

Therefore, the probability that no student solves the problem is:

P(no student solves the problem) = (1/2) x (2/3) x (3/4) = 1/4

Probability that At Least One Student Solves the Problem

Now, we can use the formula we derived earlier to find the probability that at least one student solves the problem:

P(at least one student solves the problem) = 1 - P(no student solves the problem) = 1 - 1/4 = 3/4

Therefore, the correct answer is option D (3/4).

Probability of Solving the Problem
The probability of solving the problem by the first student is 1/2, by the second student is 1/3, and by the third student is 1/4.

Probability That the Problem Will Not be Solved
The probability that the problem will not be solved by any of the students is given by the product of the probabilities that each of them does not solve it. So, the probability that the problem will not be solved is:

P(not solved) = (1 - 1/2) × (1 - 1/3) × (1 - 1/4)
P(not solved) = 1/4

Probability That the Problem Will Be Solved At Least by One
The probability that the problem will be solved at least by one student is the complement of the probability that it will not be solved by any of them. So, the probability that the problem will be solved at least by one student is:

P(solved) = 1 - P(not solved)
P(solved) = 1 - 1/4
P(solved) = 3/4

Therefore, the probability that the problem will be solved at least by one student is 3/4. Hence, option 'D' is the correct answer.