The probability of selection of three candidates A,B and C in an organization is `(2)/(5),(5)/(6)and(4)/(7)` respectively . Find the probability that at least one of them get selected.

Options

A. `(107)/(210)`
B. `(97)/(210)`
C. `(67)/(70)`
D. `(63)/(70)`

Answer

To find the probability that at least one of the candidates A, B, or C gets selected, we can use the complementary probability approach. This means we first calculate the probability that none of them gets selected, and then subtract that from 1.

### Step-by-Step Solution:

1. **Identify the probabilities of selection:**
- Probability of A being selected, \( P(A) = \frac{2}{5} \)
- Probability of B being selected, \( P(B) = \frac{5}{6} \)
- Probability of C being selected, \( P(C) = \frac{4}{7} \)

2. **Calculate the probabilities of non-selection:**
- Probability of A not being selected, \( P(A') = 1 - P(A) = 1 - \frac{2}{5} = \frac{3}{5} \)
- Probability of B not being selected, \( P(B') = 1 - P(B) = 1 - \frac{5}{6} = \frac{1}{6} \)
- Probability of C not being selected, \( P(C') = 1 - P(C) = 1 - \frac{4}{7} = \frac{3}{7} \)

3. **Calculate the probability that none of them gets selected:**
- The probability that none of A, B, or C gets selected is given by the product of their individual non-selection probabilities:
\[ P(A' \cap B' \cap C') = P(A') \times P(B') \times P(C') = \frac{3}{5} \times \frac{1}{6} \times \frac{3}{7} \]

4. **Perform the multiplication:**
\[ P(A' \cap B' \cap C') = \frac{3 \times 1 \times 3}{5 \times 6 \times 7} = \frac{9}{210} \]

5. **Simplify the fraction:**
\[ \frac{9}{210} = \frac{3}{70} \]

6. **Calculate the probability that at least one gets selected:**
- The probability that at least one of them gets selected is:
\[ P(\text{at least one selected}) = 1 - P(A' \cap B' \cap C') = 1 - \frac{3}{70} \]
\[ = \frac{70 - 3}{70} = \frac{67}{70} \]

### Final Answer:
The probability that at least one of the candidates A, B, or C gets selected is \( \frac{67}{70} \).