Question

Ten finalists are competing in a race at the Canada Games.
a) In how many different orders can the competitors finish the race?
b) How many ways could the gold, silver, and bronze medals be awarded?
c) One of the finalists is a friend from your home town. How many of the possible finishes would include your friend winning a medal?
d) How many possible finishes would leave your friend out of the medal standings?
e) Suppose one of the competitors is injured and cannot finish the race.
How does that affect your previous answers?
f) How would the competitor's injury affect your friend's chances of winning a medal? Explain your reasoning.
What assumptions have you made?

Answer 1

a) there are 10 total people. for first place, there are 10 possible people. for second place, there are 9 possible people left. and so on until the last place. so it ends up being 10! (10 factorial, not just 10)

b) repeat the logic from part a), but for only the top three places instead of all 10 places

c) consider your friend wins first place. second place therefore has 9 possible people left, and third place has 8 possible people left. there are therefore 9*8 possibilities where the friend ends up in first place. using the same logic, there are also 9*8 possibilities where the friend could end up in second and another 9*8 where the friend ends up in third place. so in all, 3 * 9 * 8

d) the solution for part a) gives the total # of possibilities. the solution for part c) gives the # of possibilities where the friend *does* end up in the medal standings. so for the # of possibilities where your friend **doesn't** end up in the medal standings, simply subtract (solution from part a - solution from part c)

e) if one person cannot finish the race, there are only 9 total people to choose from. generally speaking, the number of possibilities will decrease since there are fewer people to consider now. if you wanted to be thorough w/ this answer, you could re-calculate parts a-d with 9 people instead of 10.

f) logically, if a competitor couldn't finish the race, and all other things being equal, there would be a higher probability that any individual would place in the top 3. you could show the calculations or simply speak more generally about the number of possible permutations of 10 people vs. 9 people.