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POprobability n choose k
 

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    <p>I tried to solve this TopCoder problem: <a href="http://community.topcoder.com/stat?c=problem_statement&amp;pm=10863&amp;rd=14150" rel="nofollow">http://community.topcoder.com/stat?c=problem_statement&amp;pm=10863&amp;rd=14150</a></p> <p>But, my solution is not good, and I don't understand why.</p> <p>I understood the solution given there (down page: look for LotteryPyaterochka): <a href="http://apps.topcoder.com/wiki/display/tc/SRM+466" rel="nofollow">http://apps.topcoder.com/wiki/display/tc/SRM+466</a></p> <p>So, to sum up my problem:</p> <p>We are playing a special kind of lottery:</p> <p>Each ticket in this lottery is a rectangular grid with N rows and 5 columns, where each cell contains an integer between 1 and 5*N, inclusive. All integers within a single ticket are distinct. </p> <p>The lottery organizers randomly choose 5 distinct integers, each between 1 and 5*N, inclusive. Each possible subset of 5 integers has the same probability of being chosen. These integers are called the winning numbers. A ticket is considered a winner if and only if it has a row which contains at least 3 winning numbers.</p> <p>We want to know the number of winning ticket (thus, having at least 3 winning number in the same row)</p> <p>So, I stuck in the following step:</p> <p>number of ways of choosing the 5 numbers which appear in the 'winning row'.</p> <p>The topCoder solution says: </p> <blockquote> <p>(#ways of choosing the 5 numbers which appear in the 'winning row') = </p> <p>(#ways of choosing the x winning numbers which appear in the 'winning row') * (#ways of choosing 5-x 'non-winning numbers') = </p> <p>(5 choose x) * ((5N-5) choose (5-x))</p> <p>Since the number of winning numbers in this row is at least 3, x can be 3 or 4 or 5. So, we have (#ways of choosing the 5 numbers which appear in the 'winning row') = </p> <p>(5 choose 3) * ((5N-5) choose 2) + (5 choose 4) * ((5N-5) choose 1) + (5 choose 5) * ((5N-5) choose 0))</p> </blockquote> <p>And what I say:</p> <blockquote> <p>(#ways of choosing the 5 numbers which appear in the 'winning row') = </p> <p>(3 number among the 5 winning number) * (2 numbers to complete the row to choose among the 5N-5 non winning number + 2 winning number non chosen before) =</p> <p>(5N choose 3) * ((5N-3)choose 2)</p> </blockquote> <p>For N = 10 my method give: (5 choose 3)*(47 choose 2) = 10810</p> <p>And the topcoder method give: ((5 choose 3)<em>(45 choose 2) + (5 choose 4)</em>(45 choose 1) + (5 choose 5)*(45 choose 0)) = 10126</p> <p>Why is my method wrong ?</p> <p>Thanks</p>
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