Final answer:
The exponential generating function for arranging letters from HAWAII accounts for repetitions of 'A' and 'I' and is determined by multiplying the individual generating functions of each letter.
Step-by-step explanation:
To find the exponential generating function for the number of ways to arrange n letters from the word HAWAII, we consider the repetition of letters. The letter 'A' appears twice, and the letter 'I' appears twice as well, while the letters 'H' and 'W' appear once each.
We write the exponential generating function for a letter that appears 'r' times as (1 + x/1! + (x^2/2!) + ... + (x^r/r!)). For the letters 'A' and 'I', this will be (1 + x/1! + (x^2/2!)). For 'H' and 'W', with no repetition, it is simply (1 + x/1!).
The complete generating function is obtained by multiplying the individual generating functions for each letter:
(1 + x/1!) * (1 + x/1! + (x^2/2!))^2 * (1 + x/1!)^2
Expanding this product gives the exponential generating function for arranging the letters in HAWAII.