Distribution into Bins
If 10 people each flip a coin 3 times, is it possible for each of the 10 people to get a different sequence of Heads and Tails?
If 10 people each divide 4 balls — colored blue, green, orange, and purple — into 2 groups, is it possible for each of the 10 people to choose a different division?
Note: Each group must contain at least one ball. The groups are not labeled; a group is determined only by the balls it contains. Two divisions are considered different if there is a pair of balls that are both in the same group in one division but are in different groups in the other.
If 10 people each write 6 as a sum of non-increasing positive integers, is it possible for each of the 10 people to choose a different sum?
Note: The single number 6 is considered to be a sum, even though there is no sign.
A sum of non-increasing positive integers is a sum whose terms do not increase from left to right (though they may remain constant). For example, is a sum of non-increasing positive integers, since but is not, since
If 15 people each write down a positive integer with fewer than 4 digits whose digits add up to 4, is it possible for each of the 15 people to write a different number?
4 friends visit an ice cream parlor that offers vanilla, chocolate, and strawberry ice cream. How many ways are there for the 4 friends to each order a different flavor of ice cream?
Note: Assume each person only orders a single flavor.