How many different ways can the letters of "pollution" be arranged? The number of different ways that the letters of "pollution" can be arranged is (Simplify your answer.)

To find the number of different ways to arrange the letters in "pollution," we first note that there are 8 letters total. However, the letter "o" appears twice, and the letter "l" also appears twice. The formula for arrangements of letters, where some letters are repeated, is given by: \[ \frac{n!}{p_1! \times p_2!} \] where \(n\) is the total number of letters, and \(p_1, p_2, \ldots\) are the frequencies of the repeated letters. For "pollution," we have: - Total letters (n) = 8 - Repeated letters: "l" appears twice, "o" appears twice Therefore, the number of arrangements is: \[ \frac{8!}{2! \times 2!} = \frac{40320}{4} = 10080 \] So, the number of different ways that the letters of "pollution" can be arranged is 10,080. However, if you still want to see the breakdown of this number into a more fun context, consider this: that's enough combinations to throw a grand party where every guest could have their name in a different arrangement for multiple weeks!