5. Pete's Pockets. Can Pete distribute 44 coins into 10 pockets so that the number of coins in each pocket is different?
5. Let's "order" Petya's pockets in ascending order. Then, the smallest pocket may not contain any coins, the next one must contain at least 1 coin, and so on. Finally, in the last, tenth pocket, there must be at least 9 coins. Therefore, in all 10 pockets, there will be at least $1+2+\ldots+9=45$ coins. Answer: he will not be able to.