Question

Десять человек сдавали экзамен. они вытягивали билеты наугад по очереди по одному из 10 билетов, лежащих на столе, причем каждый вытягивал билет из оставшихся. один знал ответы ко всем 10 билетам, один - к билетам № 1.2.3,...9, один - к билетам 1, 2. 3,...8, и т.д. , один только к билету № 1. Могут ли ровно 5 человек вытянуть билеты, на которые не знают ответы? нужно решение, а не ответ

Answer 1

Могут. Пусть n-й студент знает ответы к билетам от 1 до n, тогда возможен такой расклад:
1-й вытягивает 10-й билет (не знает)
2-й вытягивает 9-й билет (не знает)
3-й вытягивает 8-й билет (не знает)
4-й вытягивает 7-й билет (не знает)
5-й вытягивает 6-й билет (не знает)
6-й вытягивает 5-й билет (знает)
7-й вытягивает 4-й билет (знает)
8-й вытягивает 3-й билет (знает)
9-й вытягивает 2-й билет (знает)
10-й вытягивает 1-й билет (знает)
Ровно 5 человек не знают ответ на свой билет, что и требовалось.

Answer 2

Problem Analysis

We have ten people who are taking an exam. They are drawing tickets randomly, one at a time, from a set of ten tickets placed on a table. Each person draws a ticket from the remaining ones. One person knows the answers to all ten tickets, one person knows the answers to tickets 1, 2, 3, ..., 9, one person knows the answers to tickets 1, 2, 3, ..., 8, and so on. One person only knows the answer to ticket number 1. The question is whether exactly five people can draw tickets for which they do not know the answers.

Solution

To determine if exactly five people can draw tickets for which they do not know the answers, let's analyze the scenario step by step.

1. The person who knows the answers to all ten tickets will draw a ticket first. They will draw any ticket since they know the answers to all of them.

2. The person who knows the answers to tickets 1, 2, 3, ..., 9 will draw a ticket next. They will draw any ticket except ticket number 1 since they already know the answer to that ticket.

3. The person who knows the answers to tickets 1, 2, 3, ..., 8 will draw a ticket next. They will draw any ticket except tickets number 1 and 2 since they already know the answers to those tickets.

4. Following the same pattern, each subsequent person will draw a ticket, excluding the tickets they already know the answers to.

Now, let's analyze the situation:

- The person who knows the answers to all ten tickets will draw a ticket first. They have no restrictions on which ticket to choose.

- The person who knows the answers to tickets 1, 2, 3, ..., 9 will draw a ticket next. They cannot draw ticket number 1 since they already know the answer to it. Therefore, they have nine options to choose from.

- The person who knows the answers to tickets 1, 2, 3, ..., 8 will draw a ticket next. They cannot draw tickets number 1 and 2 since they already know the answers to them. Therefore, they have eight options to choose from.

- Following the same pattern, each subsequent person will have one less option to choose from.

Based on this analysis, we can see that the number of available tickets decreases by one for each subsequent person. Therefore, it is not possible for exactly five people to draw tickets for which they do not know the answers. The maximum number of people who can draw tickets for which they do not know the answers is four.

Therefore, it is not possible for exactly five people to draw tickets for which they do not know the answers.

Conclusion: It is not possible for exactly five people to draw tickets for which they do not know the answers. The maximum number of people who can draw such tickets is four.