Question

The English Premier League has 20 teams, throughout the season, each team plays each of the other teams once at home and once away. (e.g. Everton plays Liverpool at Anfield, which is Liverpool’s home field, and Everton plays Liverpool at Goodison Park, which is Everton’s home field.) Teams are awarded 3 points for a win, 1 point for a draw, and 0 points for a loss

(a) How many combinations of wins, losses, and draws are possible for a team throughout the season, regardless of the order?

(b) How many ways can a team win 6 out of 10 games? Here, order matters, and you are only considering those 10 games.

Answer 1

Note: The final answers are highlighted in colour.

a)

Total of wins, losses and draws= no of matches played by each team= 19*2 =38

Therefore we need to find out the combinations of 3 numbers that add up to 38

We can see this problem as putting 38 balls in a line and using 2 stiks to separate them into 3 groups. The number of places we can put those 2 sticks gives us the number of possible combinations.

Otherwise,

Formula to split n items into k groups is

The combinations of Wins, Losses and Draws = = 741

b) How many ways can a team win 6 out of 10 games

First , we need to select the 6 winning games out of 10.

Number of ways to select 6 games= 10 C₆=210

Remaining 4 games can be either draws or losses.

Possible combinations= 2*2*2*2= 16

Therefore total number of ways a team can win 6 out of 10 games= 210*16= 3360