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)
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.