On the podium (permutations)
In a speed skating race with 8 competitors, how many ways can gold, silver, and bronze medals be awarded?
P(8,3) = 8×7×6 = 336
The USA men's hockey team has 15 players. How many ways can a coach choose 6 players for a power play unit?
C(15,6) = 5,005
Simplify: 6! / 4!
6×5 = 30
An Olympic village has 5 different restaurants. How many ways can an athlete visit at least 1 restaurant over the course of the games if they visit each one at most once?
2⁵ − 1 = 31
What is 0! equal to, and why does it matter in combinations?
1 — It ensures C(n,0) = 1, meaning there is exactly one way to choose nothing.
5 bobsled teams are competing in the finals. How many different ways can 1st, 2nd, and 3rd place be arranged?
P(5,3) = 5×4×3 = 60
A curling federation has 10 teams. How many ways can 4 teams be selected to compete in an exhibition match (order doesn't matter)?
C(10,4) = 210
A luge team wants to arrange 7 different training drills in a day. How many possible schedules are there?
7! = 5,040
A Winter Olympics broadcast features 8 events. A viewer records exactly 3 of them. How many combinations of recordings are possible?
C(8,3) = 56
A team of 9 athletes must stand in a line for a photo. How many arrangements are there if two specific athletes refuse to stand next to each other?
Total: 9! = 362,880.
Together: 2 × 8! = 80,640.
Answer: 362,880 − 80,640 = 282,240
A figure skating competition has 10 skaters. How many ways can all 10 skaters be ranked from 1st to 10th?
10! = 3,628,800
A biathlon squad of 20 athletes needs to send 3 athletes to a training camp. How many different groups can be formed?
C(20,3) = 1,140
Evaluate: (8! / (3! × 5!))
C(8,3) = 56
An ice dance competition has 6 pairs. How many ways can the judges rank all 6 pairs from best to worst?
6! = 720
A Winter Olympics schedule has 12 events. How many ways can a fan choose a morning group of 4 events and an afternoon group of 4 events from the remaining 8?
C(12,4) × C(8,4) = 495 × 70 = 34,650
A freestyle ski event has 12 athletes. How many ways can the top 4 finishing positions be filled?
P(12,4) = 12×11×10×9 = 11,880
From a roster of 12 cross-country skiers (7 men, 5 women), how many ways can a mixed relay team of 2 men and 2 women be chosen?
C(7,2) × C(5,2) = 21 × 10 = 210
How many ways can 4 identical gold medals, 3 identical silver medals, and 2 identical bronze medals be arranged in a display case?
9! / (4! × 3! × 2!) = 1,260
A Nordic combined event has 10 athletes in the ski jumping phase and 8 in the cross-country phase. 5 athletes participate in both. How many unique athletes are competing?
10 + 8 − 5 = 13 (Inclusion-Exclusion)
Using the formula C(n,r) = n! / (r!(n−r)!), prove that C(n,r) = C(n, n−r). Why does this make intuitive sense for Olympic team selection?
Substituting (n−r) for r gives the same formula. Intuitively, choosing 3 athletes to go to the Olympics is the same as choosing (n−3) athletes to stay home.
In a short track relay, a team of 5 skaters must skate in a specific order. How many different skating orders are possible if the team captain must go first?
1 × 4! = 24
A Winter Olympics committee must choose 5 sports out of 9 proposed sports to add to the program. However, if alpine skiing is chosen, snowboard cross must also be chosen. How many valid selections exist?
Total C(9,5) = 126.
Subtract invalid (alpine chosen, snowboard not): C(7,4) = 35.
Answer: 126 − 35 = 91
A ski jump scoreboard shows 9 scores. How many ways can they be arranged if the top 3 scores must always stay in the top 3 positions (in any order among themselves)?
3! × 6! = 6 × 720 = 4,320
In a round-robin hockey tournament with 6 teams, every team plays every other team exactly once. How many total games are played?
C(6,2) = 15
A figure skating event has 12 competitors. Medals go to 1st, 2nd, and 3rd. How many times greater is the number of ways to assign medals (order matters) vs. simply choosing the 3 medalists (order doesn't matter)?
P(12,3) / C(12,3) = 1320 / 220 = 6 (which equals 3!, the number of ways to arrange the 3 winners)