Q:

How many ways can I choose to eat a waffle and/or pancake in addition to my breakfast so that on one or more days I do neither?

The question goes like this: 

Each day, in addition to my breakfast I have the choice of eating a waffle, and/or eating a pancake. How many ways can I do this in a week so that on one or more days I do neither? 

I'm pretty bad with combinatorics, so I've been trying to do this in the case of 1 day, 2 days, 3 days, etc. up to a week, and then trying to see a pattern. It hasn't been working, and I don't know how to get this for seven days. Thank you for your help.

A:

An often useful skill is to count its complement (opposite) instead.
For example, the complement of (on one or more days event $X$ happens) is actually (on zero days even $X$ happens).
Each day, you have 4 choices:

breakfast
breakfast + waffle
breakfast + pancake
breakfast + waffle + pancake

Let there be $n$ days.
If there are no other restrictions, the number of choices would be $4^n$.
The number of ways to (do neither on zero days) is actually $3^n$.
Therefore, the general answer would be $4^n-3^n$.
In this case, where $n=7$, the answer is $4^7-3^7 = 14197$.

