Final answer:
You can make 1,260 different 9-letter words with the letters in 'aaaddeeee' by using the formula for permutations of a multiset and dividing 9! by the factorial of the count of each unique letter.
Step-by-step explanation:
The question asks how many 9-letter words can be formed using the letters in the word aaaddeeee. This is a problem of combinatorics, a field of mathematics that deals with counting combinations and permutations of sets. Given the repetitive letters in the word, we have 3 'a's, 2 'd's, and 4 'e's. This type of problem can be solved by using the formula for permutations of a multiset:
P(n; n1, n2, ..., nk) = n! / (n1! * n2! * ... * nk!),
where n is the total number of letters to arrange, and ni is the factorial of the number of occurrences of the ith unique letter. In this case, the formula would be: 9! / (3! * 2! * 4!) = 362880 / (6 * 2 * 24) = 1260.
Therefore, there are 1,260 different 9-letter words that can be formed.