Here we need to find the difference between the number of elements in the sets $B \bigcap C - A\bigcap B\bigcap C$ because we need to find only those numbers which are the multiples of 7 and 8 but not of 6. Now,
$\left | B\bigcap C \right | = \left \lfloor \frac{1000}{LCM (7,8)} \right \rfloor = \left \lfloor \frac{1000}{56} \right \rfloor = $ 17
$\left | A \bigcap B\bigcap C \right | = \left \lfloor \frac{1000}{LCM (6,7,8)} \right \rfloor = \left \lfloor \frac{1000}{168} \right \rfloor =$ 5
Hence elements in the shaded region = 12