Problem B. 4747. (November 2015)
B. 4747. In a certain lottery game, 6 numbers are drawn every week, out of the numbers 1 to 45. The draw of the first week of this year produced surprising results, since five consecutive numbers appeared. The numbers drawn were 37, 38, 39, 40, 41, 45. The news spread fast in the press. The question arises whether the excitement was justified: are these numbers so special? Let a number sequence be called perfect if it consists of six consecutive numbers, and nearly perfect if exactly five numbers out of the six are consecutive. How many perfect and nearly perfect combinations are there? Considering that the lottery game has been played for 26 years, and there have been 1227 weekly draws so far, what is the probability that during a time interval of this length at least one perfect or nearly perfect sequence of numbers is drawn?
Proposed by M. E. Gáspár, Budapest
Deadline expired on December 10, 2015.
184 students sent a solution. 3 points: 87 students. 2 points: 64 students. 1 point: 22 students. 0 point: 7 students. Unfair, not evaluated: 4 solutions.