Well consider the same question but with few differences:

4 options and you have to choose 3 only and you have 3 option boxes and the same rule set where box 3 has to have the same or less number of options as box2

box1 box2 box3

1. 3 0 0

2. 2 1 0

3. 1 1 1

Now then. Lets call the 4 options a,b,c, and d. Scenario 1 would have 4C3 possibilites only which is 4(there are 4 ways to choose 3 options I am assuming the order in these boxes do not matter)

Option 2 would have 4C2*2C1 possibilities which is 12(The theory here is that you choose 2 in the first box and then from the remaining two you have only one option to choose from)

ab c

ab d

ac b

ac d

and so on...

Now the final one to consider would be each box has one option. Which would be 4C1*3C1*2C1. This is my first post by the way so sorry if this is confusing.