In how many different ways (counting the orders) can the San Diego Chargers score exactly 10 points in a football game? Points are scored as follows: A safety scores 2 points, a conversion scores 2 points, a field goal 3 points, a touch down 6 points, and a point after touch down (PAT) 1 point. However, a PAT or a 2-point conversion can only be scored after a touch down.)
6+2 + 2 (conversion+safety)
6+1 + 3
6+2+2 (two safeties)
3+3+2+2
2+2+2+2+2
To find the number of different ways the San Diego Chargers can score exactly 10 points in a football game, we can consider all possible combinations of scoring events.
Let's break down the different ways the Chargers can score 10 points:
1. Safety (2 points): It is not possible to score 10 points with only safeties since the minimum combination of safeties to reach 10 points would be 5 safeties (2 points each), which is more than what is required.
2. Field Goals (3 points): To reach exactly 10 points using field goals, we need a combination of 3 field goals. So, the possible combinations would be:
a. 3 field goals
3. Touchdowns (6 points): To reach exactly 10 points using touchdowns alone, we would need 2 touchdowns. However, we have additional scoring options after a touchdown, which we will consider separately.
4. Touchdowns + Point After Touchdown (PAT) (1 point): After each touchdown, we have the option to kick a point after touchdown (PAT), which provides 1 additional point. To reach exactly 10 points, we need to consider different combinations of touchdowns and PATs. We can have:
a. 1 touchdown + 0 PATs
b. 1 touchdown + 1 PAT
c. 1 touchdown + 2 PATs
5. Touchdowns + 2-Point Conversion (2 points): Instead of kicking a PAT, teams have the option to attempt a 2-point conversion after a touchdown, which provides 2 additional points. To reach exactly 10 points, we need to consider different combinations of touchdowns and 2-point conversions. We can have:
a. 1 touchdown + 0 2-point conversions
b. 1 touchdown + 1 2-point conversion
Now, let's calculate the number of combinations for each scenario:
1. Safety: Not possible to reach exactly 10 points.
2. Field Goals: Only 1 combination of 3 field goals.
a. 3 field goals: 1 combination
3. Touchdowns:
a. 2 touchdowns: 1 combination
4. Touchdowns + PAT:
a. 1 touchdown + 0 PATs: 1 combination
b. 1 touchdown + 1 PAT: 1 combination
c. 1 touchdown + 2 PATs: 1 combination
5. Touchdowns + 2-Point Conversion:
a. 1 touchdown + 0 2-point conversions: 1 combination
b. 1 touchdown + 1 2-point conversion: 1 combination
Adding up the combinations from each scenario, we have a total of 6 different ways the San Diego Chargers can score exactly 10 points in a football game.