You own 11 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? ***Be sure to leave your answer as a fraction in order to earn credit.***


To determine the probability that the 9 CDs will be arranged in alphabetical order when chosen randomly from 11 CDs, let's break the problem down step by step.

### Step 1: Total number of ways to arrange 9 CDs
You own 11 CDs, but you only want to arrange 9 of them. The number of ways to select and arrange 9 CDs from 11 can be calculated using the formula for permutations:

\[
P(11, 9) = \frac{11!}{(11 - 9)!} = \frac{11!}{2!} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3}{2}
\]

### Step 2: Number of ways to have the CDs in alphabetical order
Regardless of which 9 CDs are chosen, there is only **1 way** for them to be arranged in alphabetical order (since there’s exactly one specific alphabetical arrangement for any set of CDs).

### Step 3: Probability calculation
The probability is the ratio of favorable outcomes (1 alphabetical arrangement) to the total number of possible arrangements (all permutations of 9 CDs from 11):

\[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{P(11, 9)}
\]

So, substituting the value of \( P(11, 9) \) we calculated:

\[
P(11, 9) = 19958400
\]

Thus, the probability is:

\[
\text{Probability} = \frac{1}{19958400}
\]

This is the exact probability that the CDs will end up in alphabetical order.

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Would you like further details or have any other questions?

Here are 5 related questions for practice:
1. How would the probability change if you only arrange 8 CDs from 11?
2. What is the probability that the 9 CDs are arranged in reverse alphabetical order?
3. How many ways can you choose 5 CDs from 11 and arrange them in any order?
4. If 5 CDs must be placed in specific positions in a rack, how many different arrangements are possible?
5. What is the probability that a random arrangement of 7 CDs from 11 is alphabetical?

**Tip:** When working with probabilities involving permutations, always check if there's a unique or specific order involved to identify favorable outcomes.

You own 11 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? Be sure to leave your answer as a fraction in order to earn credit.

Calculate the probability of randomly arranging 9 CDs in alphabetical order from a set of 11. This problem uses key concepts of permutations and probability, including the formula for P(11, 9) and calculating favorable outcomes. Suitable for high school students (Grades 9-12).

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