Answer:
To determine the possible total scores a team can have at the end of a game that is less than 100 points, we can use a method called "coin-changing problem." This method involves finding all possible combinations of coins (in this case, 3-point field goals and 7-point touchdowns with one point conversions) that add up to a given value (in this case, the total score).
Using this method, we can find that the possible total scores that the team can have at the end of a game that is less than 100 points are:
- 3, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44 ,45 ,46 ,47 ,48 ,49 ,50 ,51 ,52 ,53 ,54 ,55 ,56 ,57 ,58 ,59 ,60 ,61 ,62 ,63 ,64 ,65 ,66 ,67 ,68
To determine which totals can be made more than one way and which totals can be made the most ways without listing all the possibilities manually is to use an algorithm called dynamic programming. This algorithm involves breaking down the problem into smaller sub-problems and solving them one by one.
Using dynamic programming for this problem we can find that:
- The total score of "6" can be made in two ways: (3+3) or (7-1).
- The total score of "9" can be made in two ways: (3+3+3) or (7+1).
- The total score of "10" can be made in two ways: (3+7) or (10).
- The total score of "12" can be made in three ways: (3+3+3+3) or (7+5) or (12).
- The total score of "13" can be made in three ways: (3+3+7) or (7+3+3) or (13).
- The total score of "14" can be made in three ways: (3+11) or (7+7) or (14).
- The total score of "15" can be made in four ways: (3+3+3+3+3) or (7+3+5) or (7+8) or (15).
- The total score of "16" can be made in four ways: (3+13) or (7+3+3+3) or (7+9) or (16).
- The total score of "17" can be made in five ways: (3+3+11) or (7+5+5) or (7+10) or (10+7) or (17).
- The total score of "18" can be made in six ways: (3+3+3+3+6) or (7+11) or (10+8) or (6x3) or (9x2) or(18).
- The total score of "19" can be made in seven ways: (3 + 16), (6 + 13), (9 + 10), (12 + 7), (15 + 4), (18 + 1), and (19).
- The total score of "20" can be made in eight ways: (3+17) or (6+14) or (9+11) or (12+8) or (15+5) or (18+2) or (7x2 + 6x2) or (10x2).
- The total score of "21" can be made in ten ways: (3 + 3 + 3 + 3 + 3 + 3 + 3), (3 + 3 + 7 + 8), (3 + 18), (6 + 15), (7 + 7 + 7), (7 + 14), (10 + 11), (12 + 9), (15 + 6), and (21).
- The total score of "22" can be made in twelve ways: (3+19) or (6+16) or (9+13) or (12+10) or (15+7) or (18+4) or (7x2 + 8x2) or (10x2 + 2) or (6x3 +4) or (5x3+7) or(4x3+10).
- The total score of "23" can be made in fourteen ways: (3 + 20), (6 + 17), (9 + 14), (12 + 11), (15 + 8), (18 + 5), (7 x 3 + 2 x 5), (10 x 2 + 3 x1 ), (5 x 3 +8 x1 ), (4 x3+11 x1 ), (13+10 ), (16-1 ), and 23.
- The total score of "24" can be made in seventeen ways: (3+21) or (6+18) or (9+15) or (12+12) or (15+9) or (18+6) or (7x3 + 3x2) or (10x2 + 4) or (6x4) or (5x4 + 4) or(4x4 + 8) or(11+13 )or(14+10 )or(17+7 )or(20+4 )or(24).
- The total score of "25" can be made in twenty ways: (3+22) or (6+19) or (9+16) or (12+13) or (15+10) or (18+7) or (7x3 + 2x6) or (10x2 + 5) or(5x5)+10, (5 x 3 + 10), (5 x 4 + 5), (4 x 3 + 13), (7 x 2 + 11), (8 x 3 +1), (10 x1 +15), (12 x2 +1), (14-1), and 25.
- The total score of "26" can be made in twenty-five ways: (3+23) or (6+20) or (9+17) or (12+14) or (15+11) or (18+8) or (7x3 + 5x2) or ((10x2 + 6))or((5x5)+11 )or((5 x 3 + 12))or((5 x 4 + 6))or((4 x 3 +14))or((7 x 2 + 12))or((8 x 3 +2))or((10 x1 +16))or((12 x2 +2))or((14+12))or((17+9))or((20+6))or(26).
- The total score of "27" can be made in thirty-two ways: (3+24) or (6+21) or (9+18) or (12+15) or (15+12) or (18+9) or ((7x3 + 4x2)) or ((10x2 + 7))or((5x5)+12 )or((5 x 3 + 13))or((5 x 4 + 7))or((4 x 3 +15))or((7 x 2 + 13))or((8 x 3 +3))or(10x1+17)or(12 x2 +3)or(14+13)or(17+10)or(20+7)or(23+4)or(26-1) or(5x5+2x4)+(7x2)+1 or(6x4)+(3x3)+6 or(5x3)+(4x4)+1 or(4x3)+(7x2)+5, (8x3)+3, (6*3)+(9), (15)+(12), (11*2)+(5), (9*2)+(9), and 27.
From the above results, we can see that the total scores of "21" and "22" can be made in the most ways, with ten and twelve ways, respectively. Meanwhile, some totals, such as "10", "14", "18", and "24", can be made in multiple ways.