MathNet 2026 — Part 02 Question 34

In a certain lottery, there are 60 different numbers and 6 of them are drawn without replacement. Each ticket has 6 different numbers among the 60 possible. The maximum prize, known as "goal-in-angle", is given to the player who has the ticket with the same 6 numbers that were drawn. In this lottery, there is also the "Bola-na-Trave" award. In a ball-na-trave ticket, the smallest number has no difference, in module, greater than 1 for the smallest number drawn, the second smallest number has no difference, in module, greater than 1 for the second smallest number drawn, and so on, up to the sixth lowest number. For example, suppose the goal-in-angle ticket is $\{4,7,25,48,51,60\}$. Then tickets $\{3,6,25,49,50,59\}$ and $\{5,6,25,47,50,60\}$ are ball-na-trave tickets, but the $\{3,4,6,24,47,50\}$ ticket is not ball-on-trave. It is worth remembering that a goal-in-angle ticket is not a ball-on-trave ticket. For the following items, consider each ticket as choosing a sequence of 6 numbers written in ascending order.
(a) Give an example of a set of 6 numbers forming a goal-in-angle ticket that has the smallest possible number of ball-na-trave tickets associated with it. How many possible ball-in-trave tickets would there be for these 6 numbers?
(b) Give an example of a set of 6 numbers forming a goal-in-angle ticket that results in the greatest possible number of ball-to-trave tickets. In this case, how many ball-to-trave tickets would be possible?
(c) Consider the numbers drawn $\{2,3,8,11,14,17\}$, how many tickets are ball-na-trave associated with it?
(d) Suppose the set of numbers drawn is $\{8,10,12,14,16,18\}$. In this case, how many possible ball-to-trave tickets associated with it?