Question

1. seven guests are arriving for dinner. how many possible orders can they arrive in?

5,040!
5!
7!
49!

Explanation:

Step1: Recall Permutation of n Objects

The number of permutations (orders) of \( n \) distinct objects is given by \( n! \) (n factorial), where \( n! = n\times(n - 1)\times(n - 2)\times\cdots\times1 \). Here, we have 7 guests, so we need to find the number of permutations of 7 objects.

Step2: Apply the Factorial Formula

For \( n = 7 \), the number of possible orders is \( 7! \). Calculating \( 7! \): \( 7! = 7\times6\times5\times4\times3\times2\times1 = 5040 \), but the form of the answer is in terms of factorial, so the number of possible orders is \( 7! \).

Answer:

\( 7! \) (The option corresponding to \( 7! \))