How many different ways could each team arrive at the score of 42- 37 if the are only allowed to score 3 point field goals and 7 point touchdowns?

Question

In the Mascot Football League, the teams can score only 3- point field goals and 7-point touchdowns; so safeties or extra points are not allowed. In one contest, the SEC Mascots defeated the ACC Mascots 42- 37. In how many different ways could each team have arrived at its final score?

Answer 1

SEC Mascots scored 42.  That is some combination of 7 point touchdowns (6 or less of them) and 3 point field goals (14 of less of them). We can create a table of possible values that produce a total score of 42:

  Touchdowns        Field Goals

        0                       14

        1                       --

        2                       --

        3                        7

        4                       --

        5                       --

        6                       0

The SEC Mascots could reach their score of 42 in 3 different ways.

 

The ACC Mascots scored 37 points:

  Touchdowns        Field Goals

       0                       --

       1                       10

       2                       --

       3                       --

       4                        3

       5                       --

The ACC Mascots could have scored their 37 points in 2 ways.

Answer 2

For clarification, the restriction for scoring is:

   Touchdown is 7 points (extra point attempt waived) and field goal is 3 points;

   2 point attempts and safeties are not allowed;

 

Well for the 42 points scored by SEC Mascots we have:

  6 touchdowns OR

  3 touchdowns + 7 field goals

  

Likewise for the ACC Mascots we have:

  4 touchdowns + 3 field goals

  1 touchdown + 10 field goals

 

So there are 2 x 2 = 4 possible scoring arrangements for these two teams.

 

Just out of curiosity, what happens if the quarterback gets sacked in the end 

zone as no safeties are allowed ;-?  

Poor placekickers getting a workout