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Picking numbers (Posted on 2015-07-17) Difficulty: 3 of 5
Bernardo randomly picks 3 distinct numbers from the set (1; 2; 3; 4; 5; 6; 7; 8; 9) and arranges them in descending order to form a 3-digit number.
Silvia randomly picks 3 distinct numbers from the set (1; 2; 3; 4; 5; 6; 7; 8) and also arranges them in descending order to form a 3-digit number.

What is the probability that Bernardo's number is larger than Silvia's number?
Source: AMO

See The Solution Submitted by Ady TZIDON    
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Solution Analytical Solution (spoiler) | Comment 1 of 7
1/3 of the time Bernardo will pick a 9, and his number will be larger.

Otherwise, Bernardo and Silvia each have C(8,3) = 8*7*6/3*2*1 = 56 different possibilities.  Their numbers are equal with probability 1/56, and if they are not equal then Bernardo's is greater half of the remaining 55/56 of the time.

Altogether, the requested probability is 1/3 + (2/3)*(1/2)*(55/56) = (1/3)*(1+55/56) = 111/168 which is slightly less than 2 times out of 3 (112/168).  It would be exactly 2 out of 3 if they picked new numbers in the event of a tie. 

  Posted by Steve Herman on 2015-07-17 13:10:26
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