Question

There are $10$ pairs of shoes in a cupboard from which $4$ shoes are taken at random. Then the probability of getting:-

• A. no pair is $\frac{224}{323}$
• B. atleast one pair is $\frac{99}{323}$
• C. exactly two pairs is $\frac{3}{323}$
• D. exact;y one pair is $\frac{96}{323}$

Answer 1

Answer: (A), (C)

The correct options are
A no pair is $\frac{224}{323}$
C exactly two pairs is $\frac{3}{323}$

Probability of no pair=$\frac{20}{20}\times \frac{18}{19}\times \frac{16}{18}\times \frac{14}{17}=\frac{224}{323}$

Probability of at least one pair=1- probability of no pair=1-224/323=99/323

Probability of exactly one pair=$\frac{{{10}_C}_1+({{18}_C}_2-9)}{{{20}_C}_4}=\frac{154}{4845}$

Probability of exactly two pairs=$\frac{{{10}_C}_2}{{{20}_C}_4}=\frac{3}{323}$