Find the number of distinguishable arrangements of the letters of the word. TREDECILLION There are \( \square \) distinguishable arrangements. (Simplify your answer.)

To find the number of distinguishable arrangements of the word "TREDECILLION", we first need to count the number of letters and the frequency of each letter. The letters in "TREDECILLION" are: - T: 1 - R: 1 - E: 2 - D: 1 - C: 1 - I: 2 - L: 2 - O: 1 - N: 1 We have a total of 12 letters. The formula for the number of distinguishable arrangements of letters with repeated letters is given by: \[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \] where \( n \) is the total number of letters, and \( n_1, n_2, \ldots, n_k \) are the frequencies of each distinct letter. Thus, we calculate: \[ n = 12, \] Frequencies: - E: 2 - I: 2 - L: 2 - T, R, D, C, O, N: 1 each (8 letters) Now we can plug these values into the formula: \[ \frac{12!}{2! \times 2! \times 2! \times 1!^8} \] Calculating \( 12! \): \[ 12! = 479001600 \] Calculating \( 2! \) (it's just 2, and there are three of them): \[ 2! = 2 \] Thus, the denominator is: \[ 2! \times 2! \times 2! = 2 \times 2 \times 2 = 8 \] Now we can calculate: \[ \text{Number of arrangements} = \frac{479001600}{8} = 59875200 \] Therefore, there are \( 59875200 \) distinguishable arrangements of the letters of the word "TREDECILLION".