Dan threw 3 beans bags. Each bag went in a bucket.More than one bag can go in a bucket. E.g Bucket 1 score 1, bucket 2 score 2 , bucket 3 score 3, bucket 4 score 4

Question

1. What is the highest score Dan can get?
2. Find the three ways to score 6.
3.Find three ways to score 9.
4.
What other scores can Dan get?

Answer 1

To determine Dan's possible scores based on his bean bag throws into the buckets, let's break down the information given:

Bucket Scores:

• Bucket 1: 1 point
• Bucket 2: 2 points
• Bucket 3: 3 points
• Bucket 4: 4 points

Since Dan can throw 3 bean bags and each bag can go into any bucket, we’re essentially looking for combinations of these scores.

1. Highest Score

To find the highest score Dan can achieve with 3 bean bags, we should maximize the use of the highest scoring bucket (Bucket 4).

• If all 3 beans go into Bucket 4:
  \( 3 \times 4 = 12 \)

Therefore, the highest score Dan can get is 12.

2. Ways to Score 6

We will identify combinations of throws that sum to 6.

1. (3, 3) and (0): 3rd bean bag goes into Bucket 3 (3 points) and the rest two bean bags go to Bucket 2.
   • 2 + 2 + 2 = 6
2. (2, 2): 2nd bean bag goes into Bucket 2, 3rd bean bag goes into Bucket 2.
   • 2 + 4 = 6
3. Combination of buckets:
   • 3 + 3 + 0 = 6
   • 2 + 2 + 3 +1 = 6

Thus, three distinct ways to score 6 can be:

• Bucket 3 (2), Bucket 2 (2), Bucket 1 (0)
• Bucket 2 (2), Bucket 2 (2), Bucket 2 (0)
• Bucket 4 (3), Bucket 4 (1)

3. Ways to Score 9

To find combinations that sum to 9:

1. (4, 4, 2): Bucket 4 (4 points) and Bucket 2 (2 points) can sum to 10.
2. (3, 3, 3) - Three bags in Bucket 3 will score 9.
3. (4, 4, 1) - Two bean bags in Bucket 4 and one in Bucket 1 will score 9.

Thus, three distinct ways to score 9 can be:

• [(4), (4), (2)]
• [(3), (3), (3)]
• [(4), (4), (0)]

4. Other Possible Scores

To find all possible scores Dan can achieve with 3 bags:

• 3 bean bags can go into:
  • Bucket (0 points): 0
  • Bucket (1 point): 1
  • Bucket (2 points): 2
  • Bucket (3 points): 3
  • Bucket (4 points): 4
  • Any combination these would yield different scores, which could include:
    • 1, 2, 3, 4... and so on extending till 12.

Possible scores that can be achieved using the combinations (without the need for repeats) are:

• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Conclusion

Thus Dan can score from 0 to 12. The maximum score is 12, while scores of 6 and 9 can be achieved in multiple ways.