Determine the probability that arranging your 6 favourite books that Stevenson, Verne, and Weiss are in alphabetical order. Round answer to 3 decimal places.

To determine the probability that Stevenson, Verne, and Weiss are in alphabetical order when arranging your 6 favorite books, we can follow these steps:

1. **Identify Total Number of Arrangements**: The total number of ways to arrange 6 books is given by \(6!\) (6 factorial).
   \[
   6! = 720
   \]

2. **Identify Favorable Arrangements**: We need to count the number of favorable arrangements where Stevenson, Verne, and Weiss are in alphabetical order. The other three books can be any of the remaining books and can be in any order.

3. **Calculate Favorable Arrangements**:
   - Since Stevenson, Verne, and Weiss must be in alphabetical order, there is only 1 way to arrange these three specific books.
   - The other 3 books can be placed in any of the remaining 3 positions in \(3!\) ways (since they can be arranged among themselves).
   - Additionally, the positions for Stevenson, Verne, and Weiss within the 6 positions can be chosen in \(\binom{6}{3}\) ways (the number of ways to choose 3 positions out of 6 for these books).

   First, calculate \(\binom{6}{3}\):
   \[
   \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
   \]

   Then, the number of ways to arrange the remaining 3 books in the remaining positions:
   \[
   3! = 6
   \]

4. **Total Number of Favorable Arrangements**:
   \[
   1 \times 20 \times 6 = 120
   \]

5. **Calculate the Probability**:
   The probability \(P\) is the ratio of the number of favorable arrangements to the total number of arrangements:
   \[
   P = \frac{\text{Number of favorable arrangements}}{\text{Total number of arrangements}} = \frac{120}{720} = \frac{1}{6} \approx 0.167
   \]

Therefore, the probability that Stevenson, Verne, and Weiss are in alphabetical order when arranging the 6 books is approximately \(0.167\), rounded to three decimal places.

Would you like more details or have any questions?

Here are some related questions:

1. What is the probability that a specific pair of books are next to each other in a random arrangement of 6 books?
2. If the 6 books are all different, what is the probability that any three specified books are in a specific order?
3. How many ways can you arrange 6 books if two specific books must be on either end?
4. What is the probability of arranging 6 books such that no two specific books are adjacent?
5. What is the number of permutations of 6 books where two specific books are always together?
6. If 6 books are divided into two groups of 3, what is the number of ways to select which books go into which group?
7. How many different ways can 3 out of 6 books be selected and arranged?
8. What is the number of ways to arrange 6 books if one specific book must be in the middle position?

**Tip**: For probability questions involving permutations and combinations, always identify the total number of outcomes and the number of favorable outcomes to calculate the probability efficiently.

Learn how to calculate the probability of arranging 6 favorite books such that Stevenson, Verne, and Weiss are in alphabetical order. This problem involves permutations, combinations, and basic probability concepts. Suitable for high school students.

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