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The problem asks how many ways we can place tiles on a 1×6 rectangular grid, with 3 yellow tiles, 2 white tiles, and 1 blue tile. This is a typical combinatorics problem involving the arrangement of items with repetitions.

### Solution:

To determine the number of ways to arrange these tiles, we need to calculate the number of distinct permutations of the tiles. The formula to find the number of permutations of a multiset is:

\[
\text{Number of ways} = \frac{n!}{k_1! \times k_2! \times \dots \times k_r!}
\]

Where:
- \( n \) is the total number of items (6 tiles in total),
- \( k_1, k_2, \dots, k_r \) are the frequencies of each distinct item.

In this case:
- There are 3 yellow tiles,
- 2 white tiles,
- 1 blue tile.

So, the total number of arrangements is:

\[
\text{Number of ways} = \frac{6!}{3! \times 2! \times 1!}
\]

Let's compute this step-by-step.

\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\]
\[
3! = 3 \times 2 \times 1 = 6
\]
\[
2! = 2 \times 1 = 2
\]
\[
1! = 1
\]

Now, substituting these values into the formula:

\[
\text{Number of ways} = \frac{720}{6 \times 2 \times 1} = \frac{720}{12} = 60
\]

### Final Answer:
There are **60 ways** to arrange the tiles on the grid if there are no restrictions.

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Would you like more details or have any further questions?

Here are some related questions:
1. What if there were 4 yellow tiles, 1 white tile, and 1 blue tile?
2. How does the number of ways change if the grid was 1×7 and we added another yellow tile?
3. How do we calculate permutations when all items are distinct?
4. How can this problem be extended to a three-dimensional grid?
5. What if the tiles had additional properties, such as specific weights?

**Tip:** When solving problems with repeated items, always remember to divide by the factorial of the repeated items to avoid over-counting identical arrangements.

A 1×6 square unit rectangular grid has to be covered with six 1×1 square unit tiles. Three of the tiles are yellow, two are white, and one is blue. In how many ways can this task be done if there are no restrictions?

Learn how to calculate the number of ways to arrange three yellow, two white, and one blue tiles on a 1x6 rectangular grid. This problem explores combinatorics and the concept of permutations with repetition. Suitable for students in Grades 9-12.

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