Question

1. if there are 7 swimmers in a race, in how many different orders can they finish (assuming there are no ties)? = enter your next step here

Explanation:

Step1: Identify the problem type

This is a permutation problem where we want to find the number of ways to arrange 7 swimmers in order (since the order of finishing matters). The formula for permutations of \( n \) distinct objects is \( n! \) (n factorial), which is \( n\times(n - 1)\times(n - 2)\times\cdots\times1 \). Here, \( n = 7 \), so we need to calculate \( 7! \).

Step2: Calculate the factorial

\( 7! = 7\times6\times5\times4\times3\times2\times1 \)
First, \( 7\times6 = 42 \)
Then, \( 42\times5 = 210 \)
Next, \( 210\times4 = 840 \)
Then, \( 840\times3 = 2520 \)
Next, \( 2520\times2 = 5040 \)
Finally, \( 5040\times1 = 5040 \)

Answer:

\( 7! = 5040 \) (or directly write 5040 as the number of different orders)