IISc CDS(MTech -R)

Question

There are 12 pair of shoes, what is the probability that atleast one complete pair of shoes are present if 4 shoes are selected at random?

Comments

Hakuna Matata are there any options to the question?

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@Kushagra, no options are there.

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I think the probability should be 1

Because  if we pick 4 shoes out of 12 pairs there is always going to be 1 complete pair of shoes left.

Answer 1

Total number of ways to choose 4 shoes = $\Large\binom{24}{4}$
The probability of at least 1 complete pair of shoes is chosen = 1 - Probability of no complete pair is chosen

The probability of at least 1 complete pair of shoes is chosen $=1-\frac{24*22*20*18}{24*23*22*21}$ or $1-\frac{\binom{12}{4}*2^4}{\binom{24}{4}}$$

Comments

@srestha

1-(12C4)*2^4/24C4 also gives the same answer, please check the comment above by me, which one makes more sense to you?

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srestha ..For 1st shoe, there are 24 choices.. Then for the second shoe, only 20 choices are there and so on.  This makes sure that no complete pair is selected. 

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Hakuna Matata Both the approaches are correct. 
Choosing one shoe from a shoe pair at a time .. and repeat the process 4 times is same as 1st choosing 4 pairs of shoes and then choosing a shoe from each pair.