Question

10 teams enter a soccer tournament. In the first round teams paired in 5 pairs to play with each other. How many different pairings are possible neither the order of the pairings nor the order of teams within each pairing matters?

Answer 1

There are 945 ways to make the pairings

Explanation:

Lets pick a team. There are a total of 9 ways to select its pair. Now we have 8 teams remaining, note that the team we picked at the start doesnt matter because it will eventually be picked and the order doesnt matter. From the 8 teams remaining we select one of them and we pick a pair for it; we have 7 possibilities since we hace 7 teams remaining.

From the 6 teams remaining we take one of them and we select its partner, we now have 5 possibilities. From the 4 teams remaining we select one of them and pick 1 of the 3 teams remaining as its pair. Then, we pair the remaining teams.

This gives us a total of 9*7*5*3 = 945 ways to pair the teams.