If the number of words, with or without meaning. which can be made using all the letters of the word…

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

Now, first we will arrange letters $\_M\_A\_T\_H\_E\_M\_A\_T\_I\_$.

Here, we have $10$ gaps to fill $C \& S$.

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

$=\frac{9!}{2!·2!·2!}\times {}^{10}C_2\times 2!$

$=\frac{9\times 8\times 7\times 6!}{2!·2!}\times \frac{10\times 9}{2}$

$=5670$

Hence this is the correct option.