How many distinct permutations can be formed using the letters of the word "PROGRAMMING"? There are \( \square \) distinct permutations. (Type a whole number.)

To find the distinct permutations of the letters in "PROGRAMMING", we first count the total letters and their frequencies. The word "PROGRAMMING" has 11 letters where 'G' and 'R' appear twice, and 'M' also appears twice. The formula for distinct permutations is given by: \[ \frac{n!}{n_1! \cdot n_2! \cdot n_3!} \] where \( n \) is the total number of letters, and \( n_1, n_2, n_3 \) are the frequencies of the repeated letters. So, we have: - \( n = 11 \) (total letters) - Frequencies: - 'G' = 2 - 'R' = 2 - 'M' = 2 Calculating using the formula: \[ \text{Distinct permutations} = \frac{11!}{2! \cdot 2! \cdot 2!} = \frac{39916800}{8} = 4989600 \] Thus, there are \( 4989600 \) distinct permutations of the letters in the word "PROGRAMMING".