+489 Mary has 6 identical basil plants, and three different window sills she can put them on. How many ways are there for Mary to put the plants on the window sills?
+130466 This is analogous to filling N distinct boxes with K identical balls
If there are no restrictions as to the number of plants each sill can contain ....the number of possible ways is
C( K + N - 1, N - 1) =
C ( 6 + 3 - 1, 3 - 1) =
C (8, 2) = 28 ways
If each sill must contain at least one plant, the number of possible ways is
C( K - 1, N - 1) =
C( 6 - 1, 3 - 1) =
C(5, 2) = 10 ways
+489 By the way that is incorrect
Real Solution
Since the plants are indistinguishable, we must only count the number of plants on each window sill.
If 5 all the plants are on one window sill, there are ways to choose which window sill they are on.
If 5 plants are on one window sill and the last is on another, there are ways to choose which plants go on which window sill.
If 4 plants are on one window sill and the last two are on another, there are ways to choose which window sill they are on.
If 4 plants are on one window sill and the last two are each on one of the other windows, there are ways to choose which window the plants are on.
If 3 plants are on one window and the other plants are all on another window, there are ways to choose which window has no plants.
If 3 plants are on one window, plants on another window, and plant on the last window, there are ways to choose which plants are on which windows.
If 2 plants are on each window, there is only one way to arrange them.
In total, there are 3+6+6+3+3+6+1=28 ways to arrange the plants on the window sills.
