You own 7 CDs. You want to randomly arrange 6 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.

There is a    probability that the rack ends up in alphabetical order.

To solve this problem, we need to determine the probability that 6 randomly arranged CDs will end up in alphabetical order.

### Step 1: Total Number of Arrangements
The total number of ways to arrange 6 CDs out of 7 CDs (randomly selected) is given by the number of permutations of 6 CDs, which is:

\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\]

### Step 2: Favorable Outcome (Alphabetical Order)
There is only **one** arrangement of these 6 CDs that is in alphabetical order, because alphabetical order is a specific, unique arrangement.

### Step 3: Probability Calculation
The probability of the 6 CDs being arranged in alphabetical order is the ratio of favorable outcomes to total possible outcomes:

\[
\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{1}{720}
\]

Thus, the probability that the rack ends up in alphabetical order is:

\[
\boxed{\frac{1}{720}}
\]

Would you like a detailed explanation on how permutations were used or have any further questions?

Here are 5 related questions to further your understanding:
1. What is the probability if you arrange all 7 CDs and expect alphabetical order?
2. How would the probability change if you arranged 5 CDs from 7?
3. What is the probability that any 3 CDs selected randomly are in alphabetical order?
4. How many ways can you arrange 4 CDs out of 7 in random order?
5. Can you compute the number of ways to choose 6 CDs from 7 and then arrange them?

**Tip:** The more specific a condition, like alphabetical order, the fewer favorable outcomes there will be in probability problems.

You own 7 CDs. You want to randomly arrange 6 of them in a CD rack. What is the probability that the rack ends up in alphabetical order?

Learn how to calculate the probability of arranging 6 CDs from a set of 7 in alphabetical order. This problem involves permutations and basic probability concepts. The solution provides a clear explanation with a step-by-step breakdown of the process. Suitable for Grades 9-12.

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