The probability of solving a problem by three students X, Y and Z is `1/2, 1/3` and `1/4` respectively. the probability that the problem will be solved is :

Options

A. 44287
B. 44228
C. 44289
D. 44256

Answer

To solve the problem, we need to find the probability that at least one of the three students (X, Y, or Z) solves the problem. We can approach this by first calculating the probability that none of them solve the problem and then subtracting that from 1.

### Step-by-Step Solution:

1. **Identify the probabilities of each student solving the problem:**
- Probability that student X solves the problem, \( P(X) = \frac{1}{2} \)
- Probability that student Y solves the problem, \( P(Y) = \frac{1}{3} \)
- Probability that student Z solves the problem, \( P(Z) = \frac{1}{4} \)

2. **Calculate the probabilities of each student NOT solving the problem:**
- Probability that student X does NOT solve the problem, \( P(X') = 1 - P(X) = 1 - \frac{1}{2} = \frac{1}{2} \)
- Probability that student Y does NOT solve the problem, \( P(Y') = 1 - P(Y) = 1 - \frac{1}{3} = \frac{2}{3} \)
- Probability that student Z does NOT solve the problem, \( P(Z') = 1 - P(Z) = 1 - \frac{1}{4} = \frac{3}{4} \)

3. **Calculate the probability that none of the students solve the problem:**
- The probability that none of them solve the problem is the product of their individual probabilities of not solving it:
\[ P(X' \cap Y' \cap Z') = P(X') \times P(Y') \times P(Z') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \]

4. **Perform the multiplication:**
- First, multiply \( \frac{1}{2} \) and \( \frac{2}{3} \):
\[ \frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3} \]
- Now multiply \( \frac{1}{3} \) and \( \frac{3}{4} \):
\[ \frac{1}{3} \times \frac{3}{4} = \frac{1 \times 3}{3 \times 4} = \frac{3}{12} = \frac{1}{4} \]

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

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