Pro Football: number of ways to finish with a record of 10-4-2
As I prepare for finals this coming week, could someone tell me if I have answered the following problem correctly? Please point out any errors and I will attempt to correct the mistakes. Thanks.
The pro football season lasts 16 games. The list WWLTWWWWLWWWLLTW is the record of a team that won its first two games, lost its third, tied its fourth, etc., and finished with a record of 10-4-2, (10wins, 4 losses, 2 ties).
a) How many ways are there for a team to finish 10-4-2?
Note: use of multinomial coefficients
b) How many ways in part a) do not have consecutive losses?
First choose the ways to arrange the wins and ties. There are ways to arrange the wins and ties.
Next there are 13 ways to insert a loss into the above sequence. There are now 12 ways to insert the second loss into the sequence with one loss.
There are 11 ways to insert the third loss into the sequence of games. There are 10 ways to insert the fourth loss into the sequence of games. Therefore there are ways to play 16 games with 10 wins, 4 losses, and two ties so that there are no consecutive losses.
c) How many ways in part a) have a longest winning streak of six games?
There are 11 ways to place six consecutive wins in the sequence of 16 games. Two of these 11 either start the sequence with a win or end the sequence with a win, so we break into two cases.
First case: the winning streak happens from games 2-15. We have 4 ways to surround this winning streak with a non-win. (2 losses, 2 ties, or a loss before the win streak and a tie after, or a tie before the win streak and a loss after). These translate to the following: losses before and after ; ties before and after ; a tie before and loss after ; and a loss before and tie after . If the season starts with the win streak or ends with the win streak and has a loss on one end: or has a tie on one end: . There are therefore ways to have a longest win streak of six games.