The probabilites of `A` , `B` and `C` of solving a problem are `(1)/(6),(1)/(5) " and " (1)/(3)` respectively , What is the probability that the problem is solved ?

Options

A. `(4)/(9)`
B. `(5)/(9)`
C. `(1)/(3)`
D. none of these

Explanation

To solve the problem, we need to find the probability that at least one of A, B, or C solves the problem. We can use the complementary probability approach, which involves calculating the probability that none of them solves the problem and then subtracting that from 1.

### Step-by-Step Solution:

1. **Identify the probabilities of A, B, and C solving the problem:**
- Probability of A solving the problem, \( P(A) = \frac{1}{6} \)
- Probability of B solving the problem, \( P(B) = \frac{1}{5} \)
- Probability of C solving the problem, \( P(C) = \frac{1}{3} \)

2. **Calculate the probabilities of A, B, and C not solving the problem:**
- Probability of A not solving the problem, \( P(A') = 1 - P(A) = 1 - \frac{1}{6} = \frac{5}{6} \)
- Probability of B not solving the problem, \( P(B') = 1 - P(B) = 1 - \frac{1}{5} = \frac{4}{5} \)
- Probability of C not solving the problem, \( P(C') = 1 - P(C) = 1 - \frac{1}{3} = \frac{2}{3} \)

3. **Calculate the probability that none of them solves the problem:**
- Since A, B, and C are independent, the probability that none of them solves the problem is:
\[ P(A' \cap B' \cap C') = P(A') \times P(B') \times P(C') = \frac{5}{6} \times \frac{4}{5} \times \frac{2}{3} \]

4. **Perform the multiplication:**
- Calculate \( \frac{5}{6} \times \frac{4}{5} \times \frac{2}{3} \):
\[ = \frac{5 \times 4 \times 2}{6 \times 5 \times 3} = \frac{40}{90} = \frac{4}{9} \]

5. **Calculate the probability that at least one of them solves the problem:**
- The probability that at least one of A, B, or C solves the problem is:
\[ P(A \cup B \cup C) = 1 - P(A' \cap B' \cap C') = 1 - \frac{4}{9} = \frac{5}{9} \]

### Final Answer:
The probability that the problem is solved is \( \frac{5}{9} \).