Eight-bit strings starting with 1 or ending in 00

Consider all strings of length $8$ over the alphabet $\{0,1\}$. How many of them have first character $1$, or last two characters $00$, or both?

$160$. Inclusion–exclusion: $2^7$ start with $1$, $2^6$ end with $00$, and $2^5$ do both, so $128+64-32=160$.