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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Combination problem ##### This topic has 3 member replies ## Combination problem If n balls are placed at random into n cells, find the probability that exactly one cell remains empty. A. n/2n! B. n!/n^n C. (2n)!/n^n D. (nC2)* n!/n^n E. (nC2)* (2n)!/n^n OA: D Last edited by Anindya Madhudor on Sat Sep 07, 2013 6:48 pm; edited 1 time in total Senior | Next Rank: 100 Posts Joined 11 Jun 2013 Posted: 81 messages Followed by: 1 members Upvotes: 7 lets try with n=1 that means 1 ball with 1 cells Then no cells will remain empty.means this "exactly one cell remains empty" is an impossible event therefore Probability would be 0 putting n=1 in option A) 2 (probability >1)..hence out b) 1/1=1 out again c) 2/1=2 out again d) we can't put n=1 here as 1c2 not defined..need to further testing e) same as d now test =2.. total number exhaustive cases would be 2^2=4 favorable case 1) first cell empty and two balls gone to 2nd cells 2) two balls gone to first cell and 2nd cell is empty hence probability would be 2/2^2= 1/2 put n=2 in D and E D) 2/2^2=1/2 E) 4!/2^2=24/4=6 probability greater than 1..out Hence the answer D note: we don't need to calculate the probability for n=2 as for n=2 option E's value is greater than 1 hence wrong..Only option satisfy this is d hence right answer Senior | Next Rank: 100 Posts Joined 11 Jun 2013 Posted: 81 messages Followed by: 1 members Upvotes: 7 If we want to solve it in general way The number of ways to place n balls in n cells =n^n. There are n ways to specify the empty cell. There are n-1 ways of choosing the cell with two balls. There are nc2 ways of picking the 2 balls to go into this cell. And there are (n-2)! ways of placing the remaining n-2 balls into the n-2 cells, one ball in each cell. Hence number of ways of placing the balls such that exactly one cell is empty is n*(n-1)*(n-2)! *nc2= n!*nc2 Therefore the probability is n!*nc2/n^n Master | Next Rank: 500 Posts Joined 16 Jul 2013 Posted: 234 messages Upvotes: 9 Target GMAT Score: 700+ kinda tough question, I chose D after 10 mins • FREE GMAT Exam Know how you'd score today for$0

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