A problem in statistics is given to four students A,  B, C and D. Their chances of solving it are `(1)/(3), (1)/(4), (1)/(5)` and `(1)/(6)` respectively. What is the probability that the problem will be solved?

Options

A. `(1)/(3)`
B. `(2)/(3)`
C. `(4)/(5)`
D. None of these

Answer

To solve the problem, we need to find the probability that at least one of the students (A, B, C, or D) will solve the problem. We can do this by first calculating the probability that none of the students 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 A solves the problem, \( P(A) = \frac{1}{3} \)
- Probability that B solves the problem, \( P(B) = \frac{1}{4} \)
- Probability that C solves the problem, \( P(C) = \frac{1}{5} \)
- Probability that D solves the problem, \( P(D) = \frac{1}{6} \)

2. **Calculate the probabilities of each student NOT solving the problem:**
- Probability that A does NOT solve the problem, \( P(A') = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \)
- Probability that B does NOT solve the problem, \( P(B') = 1 - P(B) = 1 - \frac{1}{4} = \frac{3}{4} \)
- Probability that C does NOT solve the problem, \( P(C') = 1 - P(C) = 1 - \frac{1}{5} = \frac{4}{5} \)
- Probability that D does NOT solve the problem, \( P(D') = 1 - P(D) = 1 - \frac{1}{6} = \frac{5}{6} \)

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:
\[ P(\text{none}) = P(A') \times P(B') \times P(C') \times P(D') = \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \times \frac{5}{6} \]

4. **Simplify the calculation:**
- We can cancel out the common terms in the multiplication:
\[ P(\text{none}) = \frac{2 \times 3 \times 4 \times 5}{3 \times 4 \times 5 \times 6} = \frac{2}{6} = \frac{1}{3} \]

5. **Calculate the probability that at least one student solves the problem:**
- The probability that at least one student solves the problem is:
\[ P(\text{at least one}) = 1 - P(\text{none}) = 1 - \frac{1}{3} = \frac{2}{3} \]

### Final Answer:
The probability that the problem will be solved is \( \frac{2}{3} \).

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