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The problem asks how many ways the boy can arrange 5 red, 4 yellow, and 5 green marbles in a line, given that marbles of the same color are indistinguishable.

This is a classic example of a permutations problem where some objects are indistinguishable. The formula to find the number of distinct ways to arrange such items is given by the multinomial coefficient:

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

Where:
- \(n\) is the total number of objects (in this case, the total number of marbles),
- \(k_1, k_2, \dots, k_r\) are the counts of each indistinguishable object.

Here:
- Total number of marbles \(n = 5 + 4 + 5 = 14\),
- \(k_1 = 5\) (red marbles),
- \(k_2 = 4\) (yellow marbles),
- \(k_3 = 5\) (green marbles).

Thus, the number of distinct ways to arrange the marbles is:

\[
\frac{14!}{5!4!5!}
\]

Now, calculating this:

- \(14! = 87,178,291,200\),
- \(5! = 120\),
- \(4! = 24\).

So the number of distinct ways is:

\[
\frac{87,178,291,200}{(120 \times 24 \times 120)} = \frac{87,178,291,200}{345,600} = 252,252
\]

Thus, the boy can arrange the marbles in **252,252 distinct ways**.

Let me know if you would like more details on any part of the solution! Here are some related questions:

1. What if the boy had an additional 2 blue marbles?
2. How would the solution change if all the marbles were distinguishable?
3. How would you calculate the arrangements if only red marbles were distinguishable?
4. Can you generalize the formula for any number of colors and marbles?
5. How many ways can the boy arrange the marbles if he arranges them in a circle?

**Tip:** When working with factorials for large numbers, using a calculator or programming tool is very helpful to avoid manual computation errors.

A boy has 5 red, 4 yellow, and 5 green marbles. In how many ways can the boy arrange the marbles in a line if all marbles of the same color are indistinguishable?

Learn how to calculate the number of distinct ways to arrange 5 red, 4 yellow, and 5 green marbles in a line, using the multinomial coefficient formula. This problem involves key combinatorics concepts and is suitable for students in Grades 9-12.

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