Find the number of ways of arranging the letters in the word DOMINATE if
a) there are no restrictions.
We can choose the 1st letter in any of 8 ways.
We can choose the 2nd letter in any of 7 ways.
We can choose the 3rd letter in any of 6 ways.
We can choose the 4th letter in any of 5 ways.
We can choose the 5th letter in any of 4 ways.
We can choose the 6th letter in any of 3 ways.
We can choose the 7th letter either of 2 ways.
We can choose the 8th letter only 1 way.
Answer = (8)(7)(6)(5)(4)(3)(2)(1) = 8! = 40320.
b) the first letter must be a vowel.
We can choose the 1st letter in any of 4 ways, any one of {O,I,A,E}
We can choose the 2nd letter in any of 7 ways.
We can choose the 3rd letter in any of 6 ways.
We can choose the 4th letter in any of 5 ways.
We can choose the 5th letter in any of 4 ways.
We can choose the 6th letter in any of 3 ways.
We can choose the 7th letter either of 2 ways.
We can choose the 8th letter only 1 way.
Answer = (4)(7)(6)(5)(4)(3)(2)(1) = (4)(7!) = (4)(5040) = 20160.
c) the odd-numbered positions must be vowels.
We can choose the 1st letter in any of 4 ways, any one of {O,I,A,E}
We can choose the 3rd letter in any of 3 ways.
We can choose the 5th letter in either of 2 ways.
We can choose the 7th letter in only 1 way.
We can choose the 2nd letter in any of 4 ways.
We can choose the 4th letter in any of 3 ways.
We can choose the 6th letter either of 2 ways.
We can choose the 8th letter only 1 way.
Answer = (4)(3)(2)(1)(4)(3)(2)(1) = (4!)(4!) = (24)(24) = 576.
d) the last two letters must be T and E.
We can choose the 7th letter only 1 way, as T.
We can choose the 8th letter only 1 way, as E.
We can choose the 1st letter in any of 6 ways.
We can choose the 2nd letter in any of 5 ways.
We can choose the 3rd letter in any of 4 ways.
We can choose the 4th letter in any of 3 ways.
We can choose the 5th letter in either of 2 ways.
We can choose the 6th letter in only 1 way.
Answer = (1)(1)(6)(5)(4)(3)(2)(1) = 6! = 720.
[Note: I am assuming that the last two letters must be "TE" and not "ET".
It's a little unclear on that point. If it could end in "ET" the answer
would be twice as much.]
Edwin