Question

The number of $4-$digit numbers which are neither multiple of $7$ nor multiple of $3$ is

Answer 1

Let $A=$ set of all four digit integers divisible by $7.$
and $B=$ set of all four digit integers divisible by $3.$
Total number of $4$ digits numbers $=9\times 10\times 10\times 10=9000$
Let $n\left(A\right)=x$
$1001+(x-1)7=9996$
$\Rightarrow x=1286$

Let $n\left(B\right)=y$
$1002+(y-1)3=9999$
$\Rightarrow y=3000$

Let $n(A\cap B)=z$
$1008+(z-1)21=9996$
$\Rightarrow z=429$

$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)=1286+3000-429=3857$
$\Rightarrow n\left(\overline{A\cup B}\right)=9000-3857=5143$