You own 24 CDs. You want to randomly arrange 9 of them in a CD rack. What is the probability that the rack ends up in alphabetical order? The probability that the CDs are in alphabetical order is

1. Calculate the total number of ways to arrange 9 CDs out of 24. This is the number of permutations of 9 CDs, given by:

24P9 = \frac{24!}{(24-9)!} = \frac{24!}{15!}

2. For the arrangement to be in alphabetical order, there is exactly 1 way (since only one specific order is alphabetical).

3. The probability that the CDs are in alphabetical order is the number of favorable outcomes (1) divided by the total number of arrangements calculated in step 1:

\frac{1}{\frac{24!}{15!}} = \frac{15!}{24!}

4. Simplify this to get the probability that the CDs are in alphabetical order:

\frac{1}{9!}

Therefore, the final answer is: \frac{1}{9!}=\frac{1}{362880}