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