Mathematics - Permutation & Combination Question with Solution | TestHub

The total possible $4$ digit numbers are $9\times 10\times 10\times 10=9000$

Let, $X$ is the $4$ digit number divisible by $3$

$1002, 1005, \dots \dots \dots , 9999$

$a=1002, d=3 \& a_n=9999$

$9999=1002+\left(n_3-1\right)3$

$n_3=3000$

Let, $Y$ is the $4$ digit numbers divisible by $7$

$1001,1008, \dots \dots \dots , 9996$

$a=1001, d=7 \& a_n=9996$

$9996=1001+\left(n_7-1\right)7$

$n_7=1286$

Now, $4$ digit numbers divisible by both $3$ and $7$

$1008,1029,...........9996$

$a=1008, d=21, a_n=9996$

Now, $9996=1008+\left(n-1\right)21$

$n=429$

Hence,  $4$ digit numbers neither divisible by $7$ nor $3$

$=9000-1286-3000+429$

$=5143$