If the number of words, with or without meaning. which can be made...

The word 'MATHEMATICS' consists of $11$ letters including two $M ' s$, two $T ' s$ and two $A ' s$,

Now, first we will arrange letters _M_A_T_H_E_M_A_T_I_.

Here, we have $10$ gaps to fill $C \text{ and } S$.

The number of words with or without meaning can be formed from the word $MATHEMATICS$ when $C, S$ does not come together is

$= \dfrac{9 !}{2 ! \cdot 2 ! \cdot 2 !} \times C2 10 \times 2 !$

$= \dfrac{9 \times 8 \times 7 \times 6 !}{2 ! \cdot 2 !} \times \dfrac{10 \times 9}{2}$

$= 5670$

Hence this is the correct option.