The numbers 1, 2, 3, 4, 5, and 6 are to be entered into the boxes below, so that there is one number per box, and each number is used exactly once. In addition, in each row, the numbers increase from left to right, and in each column, the numbers increase from top to bottom. Find the number of possible placements. [asy] unitsize(1 cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); [/asy]
