Four darts are thrown at the target shown to the right. Assume that each of the darts land within one of the rings or within the bull’s eye. How many different point totals are possible
Suppose that four darts are thrown at the target. How many different point totals are possible if each lands on this target?
In order to determine the number of different point totals possible, we need to count the possible combinations of scores from throwing four darts at the target.
The target shown generally consists of several rings, including the bull's eye. Let's assume that the target has five different rings, each with a distinct point value:
- Inner bull (50 points)
- Outer bull (25 points)
- Inner ring (10 points)
- Middle ring (5 points)
- Outer ring (1 point)
To calculate the number of different point totals, we will start with the outer ring and determine the number of possible combinations for each ring.
Outer Ring (1 point): Each dart can either hit or miss the outer ring. Since there are four darts, there are 2^4 = 16 possible combinations of scores for this ring.
Middle Ring (5 points): Similar to the outer ring, each dart can either hit or miss the middle ring. Hence, there are 16 possible combinations for this ring.
Inner Ring (10 points): Following the same logic, there are 16 possible combinations for this ring.
Outer Bull (25 points): Each dart can either hit or miss the outer bull. Therefore, there are 16 possible combinations for this ring.
Inner Bull (50 points): Like the previous rings, each dart can either hit or miss the inner bull, resulting in 16 possible combinations.
To find the total number of point totals, we sum up the number of combinations for each ring: 16 + 16 + 16 + 16 + 16 = 80.
Therefore, there are 80 different point totals possible when throwing four darts at the target shown.
To determine the number of different point totals possible from throwing four darts at the target, we need to consider the scoring system and the possible combinations of scores.
The target typically consists of several rings with different point values, including the bullseye. Let's assume the target has five rings:
- Bullseye: 50 points
- Inner ring: 25 points
- Middle ring: 10 points
- Outer ring: 5 points
- Surrounding area: 0 points
Now, let's analyze the possible combinations of scores:
1. All four darts hit the bullseye: This is the maximum score possible, and it contributes only one unique point total of 200 (50 points each for all four darts).
2. Three darts hit the bullseye and one dart hits the inner ring: In this case, there are four possible combinations: 3 bullseye + 1 inner ring, 2 bullseye + 1 inner ring, 1 bullseye + 1 inner ring, and 1 bullseye + 2 inner rings. Each combination yields a different point total. So, there are four additional point totals: 175, 150, 125, and 100.
3. Three darts hit the bullseye and one dart hits the middle ring: Similar to the previous case, there are four possible combinations: 3 bullseye + 1 middle ring, 2 bullseye + 1 middle ring, 1 bullseye + 1 middle ring, and 1 bullseye + 2 middle rings. Each combination leads to a different point total. Thus, we have four more point totals: 160, 135, 110, and 85.
4. Three darts hit the bullseye and one dart hits the outer ring: Again, there are four possible combinations: 3 bullseye + 1 outer ring, 2 bullseye + 1 outer ring, 1 bullseye + 1 outer ring, and 1 bullseye + 2 outer rings. Each combination gives a different point total. Hence, we have four more point totals: 145, 120, 95, and 70.
5. Two darts hit the bullseye, and the other two hit the inner, middle, or outer rings: There are three possible combinations for each type of ring (inner, middle, outer), resulting in nine combinations in total. Each combination contributes a unique point total. So, we have nine extra point totals.
6. Two darts hit the bullseye, and the other two darts land outside the rings: This combination gives zero points, constituting one more point total.
In summary, we have the following point totals:
- 200, 175, 150, 125, 100 (from case 1)
- 160, 135, 110, 85 (from case 2)
- 145, 120, 95, 70 (from case 3)
- 9 additional point totals from case 5
- 1 additional point total from case 6
This yields a total of 18 different point totals possible from throwing four darts at the target.