Multiple choice

In how many different ways can four books A,B,C and D be arranged one above another in a vertical order such that the books A and B are never in continuous position?

  1. $9$
  2. $12$
  3. $14$
  4. $79$
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B Correct answer
Explanation

Total arrangements of 4 books is 4! = 24. Arrangements where A and B are together: treat (AB) as one unit, so 3! * 2! = 6 * 2 = 12. Arrangements where they are not together = 24 - 12 = 12.

AI explanation

The unrestricted number of ways to arrange the four distinct books A, B, C, and D in a vertical stack is 4!, which equals 24 arrangements. To find the arrangements where A and B are never continuous, we subtract the arrangements where A and B are always together. Treating A and B as a single block, we have 3 items to arrange, which can be done in 3! times 2! ways (the 2! accounts for swapping A and B within their block), giving 12 arrangements. Subtracting the together cases from the total, 24 minus 12 gives 12 valid arrangements.

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