Question

In a shoe rack there are 4 pairs of shoes and 4 shoes are drawn one after the other at random without replacement. Then the probability of getting at least one correct pair of shoes among the four shoes drawn is:

• A. \( \frac{8}{35} \)
• B. \( \frac{27}{35} \)
• C. \( \frac{1679}{1680} \)
• D. \( \frac{1}{1680} \)

Hint:

For probability problems involving selection without replacement, consider complementary counting: \[ P(\text{desired outcome}) = 1 - P(\text{undesired outcome}) \]

Answer 1

The Correct Option is B

Total ways to choose 4 shoes from 8: \[ \text{Total selections} = \binom{8}{4} = 70 \] Ways to select 4 shoes without forming a correct pair: \[ \text{Ways to pick one shoe from each pair} = \binom{4}{4} \times 2^4 = 8 \times 6 = 16 \] Probability of selecting 4 shoes without forming any correct pair: \[ P(\text{no correct pair}) = \frac{8}{35} \] Probability of selecting at least one correct pair: \[ P(\text{at least one pair}) = 1 - P(\text{no correct pair}) = 1 - \frac{8}{35} = \frac{27}{35} \]