Why is it easier to remember the correct response for problems with many options rather than just two? Sometimes it seems easier, at least for me, to remember an answer, when there are more than 2 options to choose from.

For example, I easily remember my pin (14 bit), but hardly the lighter switch (up or down- 1 bit) for "on".

Why is that? Or is it just a wrong impression?
The simple fact of the mater is you're talking Short Term Memory (STM) for the light switch versus Long Term Memory (LTM) for the PIN. 

From Wikipedia's definition of Long Term Memory (which is fairly accurate, if simplistic)


  Long-term memory (LTM) is memory in which associations among items are stored, as part of the theory of a dual-store memory model. According to the theory, long-term memory differs structurally and functionally from working memory or short-term memory, which ostensibly stores items for only around 20–30 seconds and can be recalled easily. This differs from the theory of the single-store retrieved context model that has no differentiation between short-term and long-term memory.


You never encode simple things like light switch positions into Long Term Memory because it would be useless; it's going to change and it's not that important (you can always check it's state readily). You pay very little attention to the light switch meaning you will likely not even keep it in Short Term Memory.

Your PIN on the other hand you have rehearsed dozens of times, tried very hard to remember, gotten some sleep (which helps the encoding process into Long Term Memory) and have built up many associations (this number gives me money, reminds me of my wallet, ect).

All forms of human memory are limited, so it's very important that we bias our memory toward important stuff; in STM you have to keep working with an item constantly to remember it; "call bob call bob call bob". But once you've elaborated on that item (I need to call bob at 3PM because he owes me money), rehersed it repeatedly, written it down (maintenance and elaborative rehearsal) it might slip into long term memory. But, since unlike your PIN, it will eventually be lost to forgetting (hopefully) since you don't continue to dwell on it, and the information stops being useful.