How many distinct distributions (x,y,z,w) of n identical marbles in 4 boxes labeled A,B,C and D are there, such that
x,y,z,w are positive integers in strictly increasing order?

Verify the validity of your formula (or set of formulas) by manual listing of all such distributions for n=18.

15 distributions were found (admittedly without brain):

1,2,3,12
1,2,4,11
1,2,5,10
1,2,6,9
1,2,7,8
1,3,4,10
1,3,5,9
1,3,6,8
1,4,5,8
1,4,6,7
2,3,4,9
2,3,5,8
2,3,6,7
2,4,5,7
3,4,5,6


Program in the Prolog language:


box(A,B,C,D) :-
       
        between(1,18,A),
        between(1,18,B),
        between(1,18,C),
        between(1,18,D),

        A < B,
        B < C,
        C < D,

        N is A+B+C+D,
        N is 18.

between(L,U,X) :-
    integer(L),integer(U),
    (   var(X) ->
        L =< U,
        between1(L,U,X)
    ;   integer(X),
        L =< X, X =< U
    ).

between1(L,U,X) :-
    (   X = L
    ;   M is L + 1, M =< U,
        between1(M,U,X)
    ).

Edited on August 16, 2017, 10:31 am

Posted by ollie on 2017-08-16 10:25:57