Question 3
$10 \mathrm{pts}$
There are 7 people competing in a foot race. In how many different orders can they finish?

Step 1 ：This problem is about finding the number of different orders in which 7 people can finish a foot race. This is a permutation problem because the order in which the people finish matters.

Step 2 ：We use the formula for permutations, which is nPr = n! / (n-r)!, where n is the total number of items, and r is the number of items to choose. In this case, n = r = 7, because all 7 people are participating in the race and can finish in any order.

Step 3 ：Substituting the values into the formula, we get 7P7 = 7! / (7-7)!. Simplifying this, we get 7! / 0!.

Step 4 ：Since the factorial of any number is the product of all positive integers less than or equal to that number, and the factorial of 0 is defined as 1, we get 7! / 1.

Step 5 ：Calculating the factorial of 7, we get 7*6*5*4*3*2*1 = 5040.

Step 6 ：So, there are \(\boxed{5040}\) different orders in which the 7 people can finish the race.