The question asks to find the number of ways to arrange 15 distinct shirts into 3 rows with 5 shirts in each row.

This is a combinatorics problem, where we need to calculate how to distribute 15 different items (shirts) into 3 groups (rows), with each group containing 5 items.

Solution approach:

1. Step 1: Select 5 shirts for the first row. We can choose 5 shirts from 15 in $\binom{15}{5}$ ways.

2. Step 2: Select 5 shirts for the second row. After selecting the first 5 shirts, we have 10 shirts remaining. We can choose 5 shirts from these 10 in $\binom{10}{5}$ ways.

3. Step 3: Select 5 shirts for the third row. Now, we are left with 5 shirts, and they all must go into the last row. This can only be done in $\binom{5}{5}$ way (which equals 1).

4. Step 4: Account for the arrangement of the 3 rows. Since the order of the rows matters (they are distinct), we need to multiply by $3!$ to account for the different ways we can arrange the 3 rows.

Formula:

$\text{Total ways} = \binom{15}{5} \times \binom{10}{5} \times \binom{5}{5} \times 3!$

Now let's compute this.

I will calculate the result.The total number of ways to arrange 15 distinct shirts into 3 rows with 5 shirts in each row is 4,540,536.

Would you like more details or have any further questions? Here are some related questions to explore:

1. How is the binomial coefficient formula derived?
2. What if the rows are identical (i.e., arrangement does not matter)?
3. How does this problem differ if the shirts are identical instead of distinct?
4. What happens if the number of rows or shirts per row changes?
5. Can this method be applied to arrange items into unequal groups?

Tip: When solving problems with combinations, make sure to check if the order of selection or arrangement matters, as this can affect whether you use permutations or combinations.