A classic counting problem is to determine the number of different ways that the letters of "occasionally" can be arranged. Find that number. If the letter that the letters will be in alphabetical order? The number of different ways that the letters of "occasionally" can be arranged is (Type an integer or a simplified fraction.)

To find the number of different arrangements of the letters in "occasionally," we start by identifying the total number of letters and the frequency of each letter. The word "occasionally" consists of 12 letters in total, and the frequency of each letter is as follows: o (1), c (2), a (2), s (1), i (1), n (1), l (1), y (1). The formula to calculate the number of unique arrangements is given by: \[ \text{Total Arrangements} = \frac{n!}{p_1! \times p_2! \times ... \times p_k!} \] Where \( n \) is the total number of letters, and \( p_1, p_2, ... p_k \) are the frequencies of each unique letter. So, we calculate: \[ n = 12 \] \[ p_{c} = 2, p_{a} = 2, p_{o} = 1, p_{s} = 1, p_{i} = 1, p_{n} = 1, p_{l} = 1, p_{y} = 1 \] Thus, the total arrangements are: \[ \frac{12!}{2! \times 2! \times 1! \times 1! \times 1! \times 1! \times 1! \times 1!} = \frac{479001600}{2 \times 2 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1} = \frac{479001600}{4} = 119750400 \] Regarding the arrangement of letters in alphabetical order, there is only one such arrangement: "aaccilnoosy". So the number of different ways that the letters of "occasionally" can be arranged is 119750400, and there is only 1 way for them to be in alphabetical order.