How many ways can 12 books be arranged on the shelf?

Answer 1

Assuming that you mean how many ways there are to order the placing them on the shelf in a left-to-right order and exclude whether they are placed the right way up, on their spine, upside down, in piles, etc, then:

12 choices for the first book, 11 for the second, and so on until 1 for the last giving:

12 × 11 × ... × 1 = 479,001,600 ways.

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For a given integer, the product of that integer and all integers less than it greater than 1 (in the above case 12 × 11 × ... × 1) is known as the factorial of that number and is written as the number followed by an exclamation mark:

3! = 3 × 2 × 1 = 6

12! = 12 × 11 × ... × 1 = 479,001,600

Zero factorial (0!) is defined to be 1.

Answer 2

12 options for the first position, 11 options for the second position, 10 for the next position, etc.; multiply everything together. This is usually written as 12! - the exclamation mark means that all numbers up to 12 are to be multiplied with one another. Scientific calculators tend to have the option to calculate that.

Answer 3

They can be arranged in 479,001,600 ways: or half that if left-to-right and right-to-left are considered the same.