A problem in mathematics is given to 4 students whose chances of solving individually are `1/2, 1/3, 1/4` and `1/5`. The probability that the problem will be solved at least by one student is

Options

A. `2/3`
B. `3/5`
C. `4/5`
D. `3/4`

Answer

To find the probability that at least one of the four students solves the problem, we can use the complementary probability approach. We will first calculate the probability that none of the students solves the problem and then subtract that from 1.

### Step-by-Step Solution:

1. **Identify the probabilities of each student solving the problem:**
- Let \( P(A) = \frac{1}{2} \) (Probability that student A solves the problem)
- Let \( P(B) = \frac{1}{3} \) (Probability that student B solves the problem)
- Let \( P(C) = \frac{1}{4} \) (Probability that student C solves the problem)
- Let \( P(D) = \frac{1}{5} \) (Probability that student D solves the problem)

2. **Calculate the probabilities of each student NOT solving the problem:**
- \( P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \)
- \( P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \)
- \( P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4} \)
- \( P(D') = 1 - P(D) = 1 - \frac{1}{5} = \frac{4}{5} \)

3. **Calculate the probability that none of the students solves the problem:**
- The probability that none of the students solves the problem is given by the product of their individual probabilities of not solving it:
\[ P(A' \cap B' \cap C' \cap D') = P(A') \times P(B') \times P(C') \times P(D') \]
Substituting the values:
\[ P(A' \cap B' \cap C' \cap D') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \]

4. **Perform the multiplication:**
- Simplifying the expression:
\[ = \frac{1 \times 2 \times 3 \times 4}{2 \times 3 \times 4 \times 5} = \frac{24}{120} = \frac{1}{5} \]

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

### Final Answer:
The probability that the problem will be solved at least by one student is \( \frac{4}{5} \).

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