My sister is playing a dart game at an arcade. She can earn 2, 3, or 6 points with each dart. She scored 24 points. How many ways could she score 24 points given that each dart thrown landed on the dart board?
2 & 3 are both factors of 6.
Meaning whenever 6 goes in a set, it can be replaced by two 3's or three 2's.
6 * 4
6 * 3; 3 * 2
6 * 3; 2 * 3
6 * 2; 3 * 4
6 * 2; 2 * 6
6 * 2; 3 * 2; 2 * 3
6 * 1; 3 * 6
6 * 1; 2 * 9
6 * 1; 3 * 4; 2 * 3
6 * 1; 3 * 2; 2 * 6
6 * 0; 3 * 8
6 * 0; 2 * 12
6 * 0; 3 * 6; 2 * 3
6 * 0; 3 * 4; 2 * 6
6 * 0; 3 * 2; 2 * 9
1 + 2 + 3 + 4 + 5 = 15 ways.
To find the number of ways your sister can score 24 points in the dart game, we can use the concept of combinations. Let's break it down step by step:
1. Determine the number of darts thrown: Since each dart thrown can earn 2, 3, or 6 points, we need to consider how many darts your sister has thrown. We can represent the number of darts as 'x'.
2. Write an equation: Based on the given information, we can write the equation 2x + 3x + 6x = 24, where '2x' represents the number of times she scored 2 points, '3x' represents the number of times she scored 3 points, and '6x' represents the number of times she scored 6 points.
3. Solve the equation: Simplifying the equation, we have 11x = 24. Dividing both sides by 11, we find that x ≈ 2.18.
4. Check for valid integers: Since the number of darts thrown should be a whole number, we need to check which values of 'x' would result in a whole number. Since 2.18 is not a whole number, we can conclude that she cannot throw a fractional number of darts. Therefore, we discard this value.
5. Find possible combinations: Now, we need to find valid combinations of scores that sum up to 24 points with whole numbers of darts thrown. Let's consider different values for 'x' and calculate the points scored in each case:
- If x = 1 (she throws one dart of each type), the total points scored would be 2 + 3 + 6 = 11. This doesn't match our target of 24 points.
- If x = 2 (she throws two darts of each type), the total points scored would be 2 * 2 + 3 * 2 + 6 * 2 = 26. This exceeds our target of 24 points.
- If x = 3 (she throws three darts of each type), the total points scored would be 2 * 3 + 3 * 3 + 6 * 3 = 39. This also exceeds our target of 24 points.
6. Determine the valid combinations: Based on our calculations above, we find that there are no valid combinations for scoring exactly 24 points using only whole numbers of darts thrown.
Therefore, there are no ways your sister can score exactly 24 points in this dart game using only whole numbers of darts thrown.