If a person throws 3 darts at a dartboard, how many possible combinations are there to get a sum of 34?
A dart board has sectors numebered from 1 to 20.
We require combinations, so order does not count.
Here's how we could proceed:
1. to avoid repetitions, we assume the smallest number is thrown first, then the next smallest, and the largest numeber.
2. There are no limit to repetitions, e.g. (7,7,20) is a valid combination.
3. Make a table of possible triplets
(1,13,20)...(7,7,20) => 7 triplets
4. then reduce the third number by one and repeat.
(1,14,19)...(7,8,19) => 7 triplets
(1,15,18)...(8,8,18) => 8 triplets
5. continue working until we get (10,12,12)...(11,11,12) => 2 triplets
6. Sum the count of all triplets, which is the correct answer.
Hint:to help you check your answer, it should be below 100 and be divisible by 9.
To find the number of possible combinations to get a sum of 34 when throwing 3 darts at a dartboard, we can use a combinatorial approach.
We can represent each dart throw as a number from 1 to 20 since that is the range of a standard dartboard. Let's call the numbers representing the dart throws as x, y, and z.
Now, we can set up the following equation to represent the sum of the dart throws:
x + y + z = 34
To find the number of combinations, we need to find the number of ways to place the '+' signs between the numbers x, y, and z.
This is known as finding the number of permutations with repetition. The formula to calculate this is:
N+r-1 C r = (N+r-1)! / [(N-1)! * r!]
Here, N represents the range of possible numbers (20 in our case), and r represents the number of variables (3 in this case).
Using this formula, the number of possible combinations is:
(20+3-1)! / [(20-1)! * 3!]
= 22!/[(19!)(3!)]
Simplifying further:
= (22*21*20!)/(19*18*3*2*1)
= (22*21)/(3*2*1)
Calculating further:
= 2 * 7
= 14
Therefore, there are 14 possible combinations to get a sum of 34 when throwing 3 darts at a dartboard.
To find the number of possible combinations to get a sum of 34 with 3 darts on a dartboard, we need to consider the different values that each dart can have and check all the possible combinations.
Let's assume the values on the dartboard are integers from 1 to 20.
To solve this problem, we can use a systematic approach:
1. Start by considering the first dart. It can have a value between 1 and 20, so we check all the possible values.
2. For each value of the first dart, we move on to the second dart. Again, it can have any value between 1 and 20.
3. For each second dart value, we determine the range of values that the third dart can take to get a sum of 34. Since it's the third dart, we can calculate it as `34 - (value of the first dart) - (value of the second dart)`. If this value is within the range of 1 to 20, it's a valid combination.
4. Repeat steps 1-3 for all possible values of the first and second darts.
To simplify the calculation, we can use nested loops to iterate through all the possible dart values:
```python
sum_combinations = 0
for first_dart in range(1, 21):
for second_dart in range(1, 21):
third_dart = 34 - first_dart - second_dart
if third_dart >= 1 and third_dart <= 20:
sum_combinations += 1
print(sum_combinations)
```
By running this code, we can find the number of possible combinations to get a sum of 34 with 3 darts on a dartboard.