### Instruction: The letters of the word `SIXTEEN' are randomly arranged.  What is the probability that the two E's are not next to each other?### Response: I will start by counting the total number of ways to arrange the letters of SIXTEEN.
There are 7 letters, but two of them are repeated, so the number of distinct arrangements is 7! / 2! = 2520.
Next, I will count the number of ways to arrange the letters so that the two E's are next to each other.
I can think of the two E's as a single unit, EE, and then arrange the remaining 5 letters around it.
There are 6 positions for EE, and then 5! ways to arrange the other letters, so the number of arrangements with EE together is 6 * 5! = 720.
Therefore, the number of arrangements with EE not together is the total minus the ones with EE together, which is 2520 - 720 = 1800.
The probability of any arrangement is 1/2520, so the probability of an arrangement with EE not together is 1800/2520 = 0.714.