Question

there are 8 books on a shelf. which expressions can you use to model the total number of ways the books can be displayed on the shelf? select all that apply. a 8! b 8·7 c 8·7·6·5·4·3·2·1 d 7!

Explanation:

Step1: Recall Permutation of n items

The number of ways to arrange \( n \) distinct items is \( n! \) (n factorial), which is defined as \( n\times(n - 1)\times(n - 2)\times\cdots\times1 \). For \( n = 8 \) books, the number of arrangements should be \( 8! \).

Step2: Analyze Option A

Option A is \( 8! \). By the definition of factorial, \( 8! = 8\times7\times6\times5\times4\times3\times2\times1 \), which represents the number of ways to arrange 8 books. So A is correct.

Step3: Analyze Option B

Option B is \( 8\cdot7 \). But the number of arrangements of 8 books is \( 8\times7\times6\times5\times4\times3\times2\times1 \), not just \( 8\times7 \). So B is incorrect.

Step4: Analyze Option C

Option C is \( 8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1 \), which is exactly the definition of \( 8! \). So C is correct.

Step5: Analyze Option D

Option D is \( 7! \). \( 7! = 7\times6\times5\times4\times3\times2\times1 \), which is not equal to \( 8! \) (since \( 8! = 8\times7! \)). So D is incorrect.

Answer:

A. \( 8! \), C. \( 8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1 \)