The probabilities of solving a problem by three students A,B and C are `1/2, 1/3 and 1/4` respectively. The probability that the problem will be solved, is

Options

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

Answer

To find the probability that the problem will be solved by at least one of the three students A, B, or C, we can use the complement rule. We will first calculate the probability that none of the students solve the problem and then subtract that from 1.

### Step-by-Step Solution:

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

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

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

4. **Perform the multiplication:**
- 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 \( \frac{1}{3} \) by \( \frac{3}{4} \):
\[ \frac{1}{3} \times \frac{3}{4} = \frac{1 \times 3}{3 \times 4} = \frac{3}{12} = \frac{1}{4} \]

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

### Final Answer:
The probability that the problem will be solved by at least one of the students A, B, or C is \( \frac{3}{4} \).

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