Susan has 10 pockets and 44 dollar bills. She wants to arrange the money that there are different number of dollars in each pocket can she do it? Explain. Solve this and show your solution

To determine if Susan can arrange the 44 dollar bills into 10 pockets so that each pocket has a different number of dollar bills, let's reason through the problem systematically.

### Step-by-Step Solution:

1. **Objective**: We want to put the 44 dollar bills into 10 pockets such that the number of bills in each pocket is different. This implies that no two pockets can have the same number of dollar bills.

2. **Smallest Arrangement**: The smallest possible distinct arrangement of dollars in 10 pockets would be giving 0 dollars to one pocket, 1 dollar to another, 2 dollars to another, and so on. The general formula for such a situation is to distribute the bills as:
   \[
   0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \text{ dollars in each pocket}.
   \]

3. **Sum of this arrangement**: To see if this works, we sum the number of dollars in all pockets:
   \[
   0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.
   \]

   The sum of the first \( n \) consecutive integers is given by:
   \[
   \text{Sum} = \frac{n(n+1)}{2},
   \]
   where \( n = 9 \) (since the largest pocket has 9 dollars):
   \[
   \text{Sum} = \frac{9(9+1)}{2} = \frac{9 \times 10}{2} = 45.
   \]
   So, the minimum distinct arrangement in 10 pockets would require 45 dollars.

4. **Conclusion**: Since Susan only has 44 dollars and the smallest arrangement with different numbers of bills in each pocket requires 45 dollars, **it is impossible** for her to arrange the bills in such a way that each pocket has a different number of bills.

### Final Answer:
Susan **cannot** arrange 44 dollar bills into 10 pockets such that each pocket has a different number of dollars, because the minimum distinct arrangement requires 45 dollars, but she only has 44.

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Do you want more details on any part of this explanation, or do you have any questions?

Here are 5 related questions to deepen understanding:
1. How would the solution change if Susan had 45 dollar bills?
2. Can Susan distribute the 44 bills if two pockets are allowed to have the same number of dollars?
3. How many bills would Susan need if she had 12 pockets instead of 10?
4. What is the formula for the sum of the first \( n \) integers, and how does it apply in general?
5. How would you distribute 44 dollar bills if the condition was that no pocket can be empty?

**Tip**: The sum of the first \( n \) integers, \( \frac{n(n+1)}{2} \), is a very useful formula for quickly determining the total of distinct distributions!

Susan has 10 pockets and 44 dollar bills. She wants to arrange the money so that there are different numbers of dollars in each pocket. Can she do it? Explain.

Explore whether Susan can arrange 44 dollar bills into 10 pockets, ensuring each pocket contains a different amount of money. This solution involves key concepts in combinatorics and arithmetic series, showing why 45 dollars are needed for a valid arrangement. Suitable for middle school students (Grades 6-8).

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