SOLUTION: What is the total number of different arrangements for the letters in the word UNITE if the first letter must be a vowel?

Any of 3 vowels U, I or E in the 1-st position from the left;   (3)

any of the remaining 4 letters in the 2-nd position;            (4)

any of the remaining 3 letters in the 3-rd position;            (3)

any of the remaining 2 letters in the 4-th position;            (2)

       the remained  1 letters in the 5-rd position.            (1)

Now, 3*4*3*2 = 72.

Answer. There are 72 possible arrangements.