(i)[2]
Find the number of distinct ways the $9$ letters of the word AGGREGATE can be arranged in a line when the first letter is R.
(ii)[2]
Find the number of distinct ways the $9$ letters of the word AGGREGATE can be arranged in a line if the $3$ letters G stay together, both letters A stay together and both letters E stay together.
(iii)[3]
The letters G, R and T are consonants, whereas A and E are vowels. Find the number of distinct ways the $9$ letters of the word AGGREGATE can be arranged in a line when vowels and consonants alternate.
(iv)[3]
Find the number of different selections of $4$ letters of the word AGGREGATE that contain exactly $2$ Gs or exactly $3$ Gs.