Subsets $S$ such that $7 \notin S $ or $2 \notin S $ 
How many subsets $S \subseteq\{1,2...10\}$ are there such that $7 \notin S $ or $2
 \notin S $?

I can't find the right way to write a formal response. I think that we should consider at least $18$ subsets, deleting once $2$ and once $7$ from the entire set. Then we should consider all the pairs and triples excluding the same two elements and so on. I'm really confused, any tips?
 A: Another possible method:
How many subsets of $\{1,2,3,4,5,6,7,8,9,10\}$ are there in total? Call this number $n$.
How many subsets contain both 2 and 7? This is the same as the number of subsets of $\{1,3,4,5,6,8,9,10\}$. Call this number $m$.
Then your solution is $n - m$.
A: Hint: 


*

*How many subsets are there containing neither $2$ nor $7$?

*How many subsets are there containing $2$ and not $7$?

*How many subsets are there containing $7$ and not $2$?


For the second one, any subset must be a union of $\{2\}$ and any subset of $\{1,3,4,5,6,8,9,10\}$. Can you say something similar for the other two?
A: Hint:
Let $S' = \lbrace 1,3,4,5,6,8,9,10\rbrace$. How many subsets does $S'$ have? Call $n$ that amount.
To any of those subsets, you can add $\lbrace 2\rbrace$ or $\lbrace 7\rbrace$ or neither to have any of the desired subsets. So you have $3n$ subsets of $S$ that don't contain $2$ or $7$.
A: Hint:
How many subsets has set $\{1,\dots,10\}$ in total?
How many of these subsets $S$ satisfy $2\in S$ and $7\in S$?
So how many do not?
