There are 6 identical black balls, 5 identical white balls and 4 identical red balls are put from left to right.
How many way(s) of arrangement is / are there

We are asked to find the number of ways to arrange 6 identical black balls, 5 identical white balls, and 4 identical red balls from left to right.

Since the balls of the same color are identical, this is a **combinatorics problem** involving the arrangement of indistinguishable objects. We can solve this using the **stars and bars** method, which is a technique for distributing indistinguishable objects into distinguishable bins.

### Total number of balls:
\[
6 \, (\text{black}) + 5 \, (\text{white}) + 4 \, (\text{red}) = 15 \, \text{balls}
\]

Now, we need to determine how to arrange 15 balls where 6 are black, 5 are white, and 4 are red. This is equivalent to finding the number of ways to arrange 15 objects where some of them are identical.

The formula for the number of ways to arrange \( n \) objects where \( a \), \( b \), and \( c \) objects are identical is given by:
\[
\frac{n!}{a!b!c!}
\]
where \( n \) is the total number of objects, and \( a \), \( b \), and \( c \) are the counts of identical objects.

Here:
- \( n = 15 \)
- \( a = 6 \) (black balls)
- \( b = 5 \) (white balls)
- \( c = 4 \) (red balls)

Substituting into the formula:
\[
\frac{15!}{6!5!4!}
\]

We can now calculate this expression.

First, compute the factorials:
- \( 15! = 1,307,674,368,000 \)
- \( 6! = 720 \)
- \( 5! = 120 \)
- \( 4! = 24 \)

Now, substitute the values:
\[
\frac{15!}{6!5!4!} = \frac{1,307,674,368,000}{720 \times 120 \times 24}
\]
\[
= \frac{1,307,674,368,000}{2,073,600} = 630
\]

Thus, the number of ways to arrange the balls is **630**.

---

Would you like further clarification on any step? Here are five related questions to consider:

1. How would the solution change if the balls were distinguishable instead of identical?
2. What if there were 3 different colors but an uneven distribution of balls?
3. How does the stars and bars method apply to problems with more than 3 categories?
4. Can you extend this method to problems involving non-ball objects, such as people or items?
5. How do factorials work, and why are they useful in combinatorics?

**Tip:** Always double-check your factorial calculations, as they grow rapidly and can affect the accuracy of combinatorial problems.

There are 6 identical black balls, 5 identical white balls and 4 identical red balls are put from left to right. How many way(s) of arrangement is / are there?

Solve the combinatorics problem of arranging 6 identical black balls, 5 identical white balls, and 4 identical red balls. Learn how to apply the stars and bars method and factorials to find the number of possible arrangements. Suitable for students in Grades 9-12.

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