counting Problem (Inclusion and Exclusion)

At Flo's shop, Flo wants to arrange 15 different plants

on five shelves for a window display.

In how many ways can she arrange them

so that each shelf has at least one, but no more than four, plants?

這問題我看解答是把問題對應到非負整數解的問題

x1 + x2 + x3 + x4 +x5 = 15 ,1<= xi <= 4 for all i = 1 to 5

1.若用GF 解的話用 EGF

2.若用 排容的話 最後答案要乘上排列數 (ie: 15!*answer)

但一般非負整數解問題都是解相同-->不同

我想問的是

這問題是相異到相異的問題

一般會想到onto(m,n)

但它卻有限制條件(onto只要箱子沒空的就可以了,不管箱子裡有幾顆球)

那有相異到相異且限制條件的問題只有1 or 2 這兩種嗎?

還是也可以用onto解???

謝謝