probability

Question

A bag contains 12  pairs of socks .four socks are picked up at random.Find the probability that there is at least one pair

Answer 1

total pairs =12 total socks =24 (12 +12)

4 socks are picked  

 prob of no pair (1 x22 x 20 x 18) /(23 x 22 x 21)  = 120/161(all are one of their pair       first one may be anything 24/24   second must not be its pair(22/23) third must not be pair of first or second    20/22   fourth must not be pair of 1st 2nd or 3rd (18/21  )

probability of at least one pair=1-120/161 =41/161

Comments

I am not getting.. anyone please explain..

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why cant be it solved like this -??

only one pair -12C₁ * 22C₂/24C₄

two pairs - 12c₂/24C₄..

TOTAL = 12C₁ * 22C₂/24C₄ + 12c₂/24C₄

can someone explain what is the problem in this approach??

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why u considering socks of same pair is different.

Answer 2

ans = one pair and other not pair  + 2 pairs

2 pairs = 12C₂

One pair and other not pair = 12C₁ * (10C₂ - 11C₁)

Thus, ans = (12C₁ * (10C₂ - 11C₁) + 12C₂) / 12C₄