A candy factory has an endless supply of red, orange, yellow, green, blue, and violet jelly beans. The factory packages the jelly beans into jars of 84 jelly beans each. As a marketing gimmick, the factory guarantees that no two jars have exactly the same number of red, orange, yellow, green, blue, and violet jelly beans. What is the maximum number of jars the factory can produce?
How many ways to put N identical objects into K different cells: .
So when I asked about many solutions to ,
that is equivalent putting 84 identical ones into six different colors.
Each solution is one way the candy factory can package the 84 jelly beans.
Suppose that an ice cream shop has 21 flavors.
It runs a special offer of a plate of 4 scoops of any combinations.
How many different offerings are there?
That is the number of ways to put 4 identical objects (scoops) into 21 (flavors) different cells.
Note that the 4 can be all different, all the same, or say two of one and two different.