Question

How many ways are there to distribute 15 district objects into 5 distinct boxes with: (i) At least three empty box. (ii) No empty box.

Answer 1

The way of partitioning given n distinct objects into k identical (unlabeled) bins is given by Sterling numbers of the second kind.    We have to put at least one object in one box.
                   {  n  } 
     S(n, k) = {      }      n>=1,   k>=1,    n>=k 
                   {  k  }

S^(k) n = S(n,k) = 1/k! ∑ kj = 0 (-1) k-j k Cj j^n


The number of ways to distribute n distinct objects into k distinct labeled boxes (at least one object in each box) is given by  k! * S(n, k).   This is because we can permute the k boxes in k! ways.  let us call these numbers as f(n, k) = k! * S(n, k)

We can compute these sterling numbers for different n and k.
  S(0,0) = 0.  S(1,1) =1,    S(n,1) = 1      S(n, n) = 1

   f(0,0) = 0    f(1,1) = 1      f(n , 1) = 1     f(n, n) = n !
   f(n, 2)  = 2^n  - 2
 
There is a recurrence relation among these numbers.

  S(n+1, k) =  k S (n, k) + S(n, k-1)

  f(n+1, k) = k [ f(n, k) + f(n , k-1) ]


Thus we can find their values. with at least one object in each of 10 boxes:

f(20,10) = - 10C1 x 1^20 + 10C 2 x 2^20 - 10C3 x 3^20 + 10C4 x 4^20

- 10C5 x 5^20 + 10C6 x 6^20 - 10C7 x 7^20

+10C8 x 8^20 - 10C9 x 9^20 + 10^20

= 21473732319740100000
= 2.1473 x 10^19

The result should actually be a multiple of 10!

For part 1 of the question:  with at least 3 empty boxes.

A number of the ways with all distinct objects being distributed in any fashion, with any number of objects in any distinct box.  = 10^20
   as for each object there are 10 ways, by putting it in any of the boxes.

Ways of 20 distinct objects being put in 10 distinct boxes, with one box empty:
    10C1  * f(20, 9)  = 10 * f(20, 9)

Ways of 20 distinct objects being put in 10 distinct boxes, with two boxes empty:
    10C2  * f(20, 8)  = 45 * f(20, 8)
 
So the answer for distribution of 20 objects in 10 boxes (distinct) with at least 3 boxes empty will be =   10^20 - 10 f(20, 9) - 45 *f(20,8)

This number will also be in the range of 10^19.