Task cycle

Greetings. At one time, I spent some time on preparing students for olympiad tasks (grades 5-6). And recently I read the famous book “ How to move Mount Fuji? ” And realized that the tasks there are mostly similar. And since today is Friday, I decided to share some interesting puzzles.

1. It is known that 40! = 81591528324 Q 897734345611269596115894272000000000. Find the digit Q (Without calculating 40! Of course) .
2. In how many ways can you accommodate 7 students in three rooms: single, double and quadruple?
3. In the bag of 24 kg of nails. How, having only scales without an arrow, measure 9 kg of nails?
4. There are 101 coins, of which 50 are fake, differing in weight by 1 gram from the real ones. Petya took one coin and for one weighing on the scales with an arrow, showing the difference on the cups in grams, she wants to determine if she is fake. Can he do it?
5. 9 numbers are placed in a circle - 4 ones and 5 zeros. Every second, the following operation is performed on the numbers: they put zero between neighboring numbers, if they are different, and one if they are equal; after that, the old numbers are erased. Can after a while all the numbers become the same?

Good luck with your decision! Today, the tasks chosen are not very complex. If anything, next Friday will be more difficult.
')
UPD : In the comments, the answers, if you want to think for yourself, for now, just freak out without comment.