Question

You own 18 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? Give your decimal answer to at least 4 digits after the 0s

Answer 1

To find the probability that 6 randomly chosen CDs from a total of 18 end up in alphabetical order when placed in a rack, we can follow these steps:

1. **Determine the number of ways to choose and arrange 6 CDs**:
- First, we must choose 6 CDs out of 18. The number of ways to choose 6 CDs from 18 is given by the combination formula, \(\binom{n}{k}\), where \(n\) is the total number of items and \(k\) is the number of items to choose.
\[
\binom{18}{6} = \frac{18!}{6!(18-6)!} = \frac{18!}{6! \cdot 12!}
\]
- This calculates to:
\[
\binom{18}{6} = 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 / (6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) = 18564
\]

- Next, for each selection of the 6 CDs, they can be arranged in any order. The total number of arrangements of 6 CDs is \(6!\):
\[
6! = 720
\]

- Therefore, the total number of ways to choose and arrange 6 CDs from 18 is:
\[
\binom{18}{6} \cdot 6! = 18564 \cdot 720 = 13365120
\]

2. **Determine the number of ways for the selected CDs to be in alphabetical order**:
- For any selection of 6 CDs, there is exactly **1 way** to arrange them in alphabetical order.

3. **Calculate the probability**:
- The probability that the arrangement is in alphabetical order is the ratio of the favorable outcomes (which is 1) to the total outcomes (which we calculated to be 13365120):
\[
P(\text{alphabetical order}) = \frac{1}{13365120}
\]

- Now, calculating this probability gives:
\[
P(\text{alphabetical order}) \approx 7.47 \times 10^{-8}
\]

4. **Convert to decimal format**:
\[
P \approx 0.0000000747
\]

Rounding this to four decimal places after the 0 gives:
\[
\boxed{0.0000}
\]

Thus the probability that the 6 randomly chosen CDs are arranged in alphabetical order is approximately \(0.0000\).