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:
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Determine the number of ways to choose and arrange 6 CDs:
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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. \[ \binom{18}{6} = \frac{18!}{6!(18-6)!} = \frac{18!}{6! \cdot 12!} \]
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This calculates to: \[ \binom{18}{6} = 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 / (6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) = 18564 \]
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Next, for each selection of the 6 CDs, they can be arranged in any order. The total number of arrangements of 6 CDs is \(6!\): \[ 6! = 720 \]
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Therefore, the total number of ways to choose and arrange 6 CDs from 18 is: \[ \binom{18}{6} \cdot 6! = 18564 \cdot 720 = 13365120 \]
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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.
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Calculate the probability:
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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): \[ P(\text{alphabetical order}) = \frac{1}{13365120} \]
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Now, calculating this probability gives: \[ P(\text{alphabetical order}) \approx 7.47 \times 10^{-8} \]
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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 \(0.0000\).
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