A

The arrangement is made in such a way that the vowels always come together.
i.e., "CHRGS(AE)".
Considering vowels as one letter, 6 different letters can be arranged in 6! ways; i.e., 6! = 720 ways.
The vowels "AE" can be arranged themselves in 2! ways; i.e.,2! = 2 ways
Therefore, required number of ways = 720 x 2 = 1440 ways.

B

There are 9 different letters in the given word "COMPLAINT", out of which there are 3 vowels and 6 consonants.
Let us mark these positions as under:
[1] [2] [3] [4] [5] [6] [7] [8] [9]
Now, 3 vowels can be placed at any of the three places out of 5 marked 1, 3, 5, 7 and 9.
Number of ways of arranging the vowels = [latex] ^{5}{P}_{3}[/latex] = 5 x 4 x 3 = 60 ways.
Also, the 6 consonants at the remaining positions may be arranged in [latex] ^{6}{P}_{6}[/latex] ways = 6! ways = 720 ways.
Therefore, required number of ways = 60 x 720 = 43200 ways.

A

There are 9 letters in the given word, out of which 4 are vowels.
In the word "CANDIDATE" we treat the vowels "AIAE" as one letter.
Thus, we have CNDDT(AIAE).
Now, we have to arrange 6 letters, out of which D occurs twice.
Therefore, number of ways of arranging these letters = [latex] \frac {6!}{2!} = \frac {720}{2} = 360 ways.[/latex]
Now, AIAE 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!} = \frac {1 \times 2 \times 3 \times 4 }{2} = 12[/latex]
Therefore, required number of words = (360 x 12) = 4320.

D

There are 6 different 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 of 3 marked 1, 3 and 5.
Number of ways of arranging the vowels = [latex] ^{3}{P}_{3}[/latex] = 3! = 6 ways.
Also, the 3 consonants can be arranged at the remaining 3 positions.
Number of ways of these arrangements = [latex] ^{3}{P}_{3} = 3! = 6[/latex] ways.
Therefore, total number of ways = 6 x 6 = 36.

A

There are 7 different letters in the word 'SERVING'.
Therefore, the number of arrangements of the seven letters of the word = Number of all permutations of 7 letters, taken 7 at a time = [latex] ^{n}{P}_{n}[/latex] = n(n - 1)(n - 2) ... (n - n + 1) = n!
Here, n = 7 then required number of ways = 7! = 5040.