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

The problem can be approached using the **pigeonhole principle**, which helps determine whether a certain distribution is possible.

### Explanation:
Susan has 10 pockets and 44 dollar bills, and she wants to distribute the bills in such a way that **each pocket contains a different number of dollars**.

Let's analyze this step by step:

1. If she is to put a different number of bills in each pocket, the number of bills in each pocket must be a distinct integer. Therefore, she needs 10 distinct integers for the number of bills in each pocket.
   
2. The smallest possible distinct integers are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These are the first 10 distinct non-negative integers.

3. The sum of these integers is:
   \[
   0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
   \]
   So, the minimum total number of bills required to fill 10 pockets with distinct numbers of bills is **45**.

4. However, Susan only has **44** bills. Since 44 is less than 45, it's **impossible** to distribute 44 bills into 10 pockets with different numbers in each pocket. One pocket would need at least 0 bills, and there would still not be enough bills for the rest of the pockets.

### Conclusion:
Susan **cannot** arrange 44 dollar bills into 10 pockets such that each pocket has a different number of dollars.

Would you like further clarification or more details? Here are five related questions for practice:

1. If Susan had 45 dollar bills, could she arrange them with different amounts in each pocket?
2. How would the distribution change if Susan had only 9 pockets instead of 10?
3. What if Susan had 11 pockets and 44 dollar bills—what would happen in this case?
4. How can the pigeonhole principle be applied to larger numbers of pockets and bills?
5. How would the solution differ if Susan allowed for some pockets to have the same number of bills?

**Tip:** The pigeonhole principle is useful in situations where you need to prove that something must repeat, especially when the number of "items" exceeds the number of "places" to put them.

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.

Explore whether Susan can distribute 44 dollar bills across 10 pockets with different amounts using the pigeonhole principle. Understand the mathematical reasoning and find out why it's impossible. Suitable for Grades 8-10.

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