12.  How many 4letter words with or without meaning, can be formed out of the letters of the word, ‘LOGARITHMS’, if repetition of letters is not allowed?  
Answer: Option C Explanation:
‘LOGARITHMS’ contains 10 different letters.

13.  In how many different ways can the letters of the word ‘MATHEMATICS’ be arranged so that the vowels always come together?  
Answer: Option C Explanation:
In the word ‘MATHEMATICS’, we treat the vowels AEAI as one letter. Thus, we have MTHMTCS (AEAI). Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.
Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Required number of words = (10080 x 12) = 120960. 
14.  In how many different ways can the letters of the word ‘OPTICAL’ be arranged so that the vowels always come together?  
Answer: Option B Explanation:
The word ‘OPTICAL’ contains 7 different letters. When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL (OIA). Now, 5 letters can be arranged in 5! = 120 ways. The vowels (OIA) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 x 6) = 720. 