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There are 'n' pair of shoes. A thief thieves 'r' pair of shoes,he is unsucceeded. Find the ways, there is no matching.
Hi Alankrita,
n pair of shoes----- n left and n right = 2n shoes
r pairs means 2r shoes.
thief picks up 2r shoes.
Let us think about how can we arrange the 2r objects selecting from 2n shoes in 2r places no is no perfect pair------- which tells thief is unsucceded considering the order of picking.
no of ways, S= 2n*(2n-2)*(2n-4)*(2n-6)*.......*(2n-4r+2)
= 2^2r * n*(n-1)*(n-2)*....*(n-2r+1) = 2^2r * n↓/(n-2r)↓
If the thief picks randomly not considering the order
then no of ways= S/(2r)↓
= 22r * n!/[(n-2r)! * (2r)!] = 22r * nC2r ----------ANS
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