Question

A football team has 20 teams, with half of them in the West division and the other half in the East division. 8 top teams advance to a playoff at the end of a season.

a. How many ways can the top 8 teams be chosen if they can be from any division?

b. How many ways can the top 8 teams be chosen if each division must chose 4 teams?

Answer 1

Solution:

Given:

A football team has 20 teams, with half of them in the West division and the other half in the East division.

That is: 10 teams from the West division and 10 teams from the the East division.

8 top teams advance to a playoff at the end of a season.

Part a) How many ways can the top 8 teams be chosen if they can be from any division?

Number of ways in which   the top 8 teams be chosen from any division = ₂₀C₈

Thus

Number of ways in which   the top 8 teams be chosen from any division = ₂₀C₈

Number of ways in which   the top 8 teams be chosen from any division = 125970

Part b) How many ways can the top 8 teams be chosen if each division must chose 4 teams?

That is: 4 teams from from the 10 West division and 4 teams from the 10 East division.

Thus

Number of ways in which the top 8 teams be chosen if each division must chose 4 teams = ₁₀C₄ * ₁₀C₄

Thus find ₁₀C₄

Thus

Number of ways in which the top 8 teams be chosen if each division must chose 4 teams = ₁₀C₄ * ₁₀C₄

Number of ways in which the top 8 teams be chosen if each division must chose 4 teams = 210 *210

Number of ways in which the top 8 teams be chosen if each division must chose 4 teams = 44100