C

There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.
Let us mark these positions as under:
(1) (2) (3) (4) (5) (6)
Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.
Number of ways of arranging the vowels = [latex]^{3}{P}_{3}[/latex] = 3! = 6.
Also, the 3 consonants can be arranged at the remaining 3 positions.
Number of ways of these arrangements = [latex]^{3}{P}_{3}[/latex] = 3! = 6.
Total number of ways = (6 x 6) = 36

A

= [latex](^{3}{P}_{3} \times ^{3}{P}_{3}) = (^{7}{C}_{2} \times ^{3}{C}_{1}) = (\frac {7 \times 6}{2 \times 1} \times 3) = 63[/latex]

C

'LOGARITHMS' contains 10 different letters.
Required number of words = Number of arrangements of 10 letters, taking 4 at a time.
= [latex]^{10}{P}_{4}[/latex]
= (10 x 9 x 8 x 7)
= 5040.

C

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.
i.e, Number of ways of arranging these letters = [latex]\frac {8!}{(2!)(2!)}[/latex] = 10080.
Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters = [latex]\frac {4!}{2!}[/latex] = 12
i.e, Required number of words = (10080 x 12) = 120960.

B

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.
i.e, Required number of ways = (120 x 6) = 720.