Distributing objects into boxes

Solve the first problem 17 problems

Marbles for distinct counts ●○○○○ 10 ptsLisa has 10 friends and 34 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble and no two friends receive the same number of marbles?
Distributing distinct problems to friends ●○○○○ 10 ptsAlex has 5 pieces of paper, each with a different math problem. In how many ways can he give these problems to his 10 friends (he can give each friend more than one problem)?
Choosing two balls from twenty ●○○○○ 10 ptsTwenty balls numbered 1 through 20 are in a bin. In how many ways can 2 balls be selected if the order in which they are drawn does not matter?
Equal votes in a village election ●●○○○ 20 ptsA small village has $n$ people. During their yearly elections, every group of three people comes up to a stage and, as a group, casts a single vote for someone in the village to be the new leader. After every possible group of three people has voted, it turns out that every person in the village received exactly the same number of votes. If $10 \leq n \leq 100$ , how many possible values of $n$ are there?
Gifts to five children with restrictions ●●○○○ 20 ptsFour types of present are available: a candy, a paperback, a dolly, and a Mercedes. Five children, Jackie, Tito, Jermaine, Marlon and Michael, each receive exactly one present. Michael and Jermaine cannot receive the same type of present, Tito and Michael cannot receive the same type of present, and Tito cannot receive a paperback. In how many ways can the presents be given?
Athletes taking someone else's ball ●●○○○ 20 ptsFive athletes came to training, each with their own ball. When leaving, each athlete took a ball that was not their own, and each ball was taken by exactly one athlete. In how many ways is this possible?
Maximum elephants with a unique distribution ●●●○○ 50 ptsOstap Bender organized an elephant giveaway in the city of Fux. 28 union members and 37 non-members showed up, and Ostap distributed the elephants so that all union members received the same number of elephants and all non-members received the same number of elephants (the two numbers may differ). It turned out that there was only one way to distribute the elephants so that all of them were given away. What is the maximum number of elephants Ostap Bender could have had? (Each person who came received at least one elephant.)
Placing plants under colored lamps ●●●○○ 50 ptsRachel has two identical basil plants and one aloe plant. She has two identical white lamps and two identical red lamps. Each plant is placed under exactly one lamp, and more than one plant may share a lamp. How many ways are there for Rachel to place her plants under her lamps?
Three piles of coins with increasing sizes ●●●○○ 50 ptsFind the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.

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