The calculation for the number of ways of putting $m$ balls in $n$ boxes with each box having at **least one ball** in Haskell:

fact 0 = 1
fact 1 = 1
fact x = x * fact (x - 1)
balls m 1 = 1
balls m n | m < n = 0
balls m n | m == n = fact(n)
balls m n = n * balls (m -1) n + n * balls (m - 1) (n - 1) |

fact 0 = 1
fact 1 = 1
fact x = x * fact (x - 1)
balls m 1 = 1
balls m n | m < n = 0
balls m n | m == n = fact(n)
balls m n = n * balls (m -1) n + n * balls (m - 1) (n - 1)

Testing:

$ ghci
GHCi, version 7.6.3: http://www.haskell.org/ghc/ :? for help
Loading package ghc-prim ... linking ... done.
Loading package integer-gmp ... linking ... done.
Loading package base ... linking ... done.
Prelude> |

$ ghci
GHCi, version 7.6.3: http://www.haskell.org/ghc/ :? for help
Loading package ghc-prim ... linking ... done.
Loading package integer-gmp ... linking ... done.
Loading package base ... linking ... done.
Prelude>

Prelude> :l balls.hs
[1 of 1] Compiling Main ( balls.hs, interpreted )
Ok, modules loaded: Main.
*Main> balls 12 3
519156
*Main> balls 50 2
1125899906842622
*Main> balls 2 2
2
*Main> balls 3 2
6
*Main> balls 4 5
0
*Main> balls 5 10
0
*Main> balls 10 5
5103000
*Main> balls 10 2
1022
*Main> balls 10 9
16329600 |

Prelude> :l balls.hs
[1 of 1] Compiling Main ( balls.hs, interpreted )
Ok, modules loaded: Main.
*Main> balls 12 3
519156
*Main> balls 50 2
1125899906842622
*Main> balls 2 2
2
*Main> balls 3 2
6
*Main> balls 4 5
0
*Main> balls 5 10
0
*Main> balls 10 5
5103000
*Main> balls 10 2
1022
*Main> balls 10 9
16329600