It’s move-in day, and Grace is admiring the beautiful facade of her new dorm, Gregory. Being
geometrically inclined, she observes that the windows form a 16 × 16 grid. Excited to beautify
the dorm even further, she decides to decorate each of the 256 windows with empty flower
boxes. Each flower box can only be put by one window and all the flower boxes will stay in
their respective window.
(a) Now, Grace is ready to plant flowers in the boxes. She decides that she wants to distribute
16 indistinguishable roses to her boxes such that no two rose-filled flower boxes are in the
same row or column. How many ways can Grace fill the flower boxes with roses?

This problem involves distributing 16 indistinguishable roses into 16 flower boxes, arranged in a 16 × 16 grid, such that no two rose-filled boxes are in the same row or column. This means Grace wants to place exactly one rose in each of 16 rows and each of 16 columns, effectively finding a way to select 16 positions in the grid where each row and each column has exactly one rose-filled box.

### Problem Interpretation:
This is a classic problem of counting the number of ways to place non-attacking rooks on a chessboard, known in combinatorics as counting the number of permutations. Specifically, we need to find how many ways we can arrange 16 roses such that each row and each column contains exactly one rose.

### Step-by-Step Solution:

1. **Permutations of Columns**: 
   The problem of distributing the roses can be thought of as choosing one unique column in each row to place a rose. Since the rows are distinct and no two roses can be placed in the same column, the number of ways to assign one column to each row is simply the number of permutations of 16 columns.

2. **Number of Permutations**: 
   The number of ways to permute 16 distinct objects (the columns) is given by the factorial of 16, written as \( 16! \).

   \[
   16! = 16 \times 15 \times 14 \times \cdots \times 2 \times 1
   \]

3. **Final Answer**: 
   The number of ways Grace can place the roses is \( 16! \).

   To calculate \( 16! \):

   \[
   16! = 20,922,789,888,000
   \]

Thus, there are **20,922,789,888,000** ways Grace can distribute the 16 roses such that no two roses are in the same row or column.

### Conclusion:
The number of ways Grace can fill the flower boxes with roses is \( 16! = 20,922,789,888,000 \).

Would you like further clarification, or details on any of the steps?

### Related Questions:
1. How many ways are there to distribute the roses if the grid were \( 8 \times 8 \) instead of \( 16 \times 16 \)?
2. What if Grace could plant any number of roses in each box, but no box could contain more than 2 roses?
3. If there were only 10 roses and they needed to be placed such that no two are in the same row or column, how many ways could Grace do this?
4. What if the grid were \( 16 \times 16 \) but Grace could only place roses in alternate columns, how many ways could she arrange the roses?
5. How would the problem change if the flower boxes were distinguishable, but the roses were distinguishable as well?

### Tip:
Remember, factorials grow very quickly! For large grids like the \( 16 \times 16 \), the number of permutations is immense. Factorial shortcuts or approximations like Stirling's approximation are often useful for such calculations.

It’s move-in day, and Grace is admiring the beautiful facade of her new dorm, Gregory. Being geometrically inclined, she observes that the windows form a 16 × 16 grid. Excited to beautify the dorm even further, she decides to decorate each of the 256 windows with empty flower boxes. Each flower box can only be put by one window and all the flower boxes will stay in their respective window. (a) Now, Grace is ready to plant flowers in the boxes. She decides that she wants to distribute 16 indistinguishable roses to her boxes such that no two rose-filled flower boxes are in the same row or column. How many ways can Grace fill the flower boxes with roses?

Explore the solution to a combinatorics problem where 16 indistinguishable roses are distributed into a 16x16 grid with no two roses in the same row or column. Learn how to apply permutation and factorial concepts to solve this unique arrangement problem. The total number of arrangements is 16! = 20,922,789,888,000.

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