The number of 3 -digit numbers, that are divisible by either 2 or 3...

We know that total number of three-digit number will be $900$,

Now let $n\left(A\right)$ be number of three-digit number which are divisible by $2$, i.e,

$A = \left\{100, 102, . . . , 998\right\}$

$⇒ n\left(A\right) = 450$

$n\left(B\right)$ be number of three-digit number which are divisible by $3$, i.e, $B = \left\{102, 105, . . . , 999\right\}$

$n\left(B\right) = 300$

Numbers divisible by both $2$ and $3$ is

\left\{102, 108, . . . 996\right\}$i.e,$150$</p><p>Numbers divisible by both$2$and$7$are \left\{112, 126, . . . , 994\right\}$ i.e., $64$ numbers.

Numbers divisible by both $3$ and $7$ is \left\{105, 126, . . . 987\right\}$i.e,$43$numbers.</p><p>Numbers divisible by$2, 3 \text{ and } 7$is \left\{126, 168, . . . , 966\right\}$ i.e, $21$ numbers.

Required number

$= 450 + 150 - 64 - 43 + 21 = 514$

Hence, $514$ three-digit number are there which are divisible by $2\text{or}3\text{but not divisible by}7$.