Disposal, disposition are frequently used without clear distinction when they mean the act or the power of disposing of something.
However, when the emphasis is upon what shall be done with money, property, or possessions, disposal tends to imply a getting rid of (as by selling, giving away, assigning to others, or destroying) and disposition,a proper or orderly distribution or utilization.
When the idea of arrangement or ordering or of making arrangements is stressed, disposition, rather than disposal, is the more accurate term.
The idiomatic phrases at one’s disposal and at (or in) one’s disposition differ in that, though both imply a placing under one’s control, the former suggests use as one sees fit and the latter, subjection to one’s direction, arrangement, or command.