Answer the following question

There are 3 different types of gems: Ruby, Sapphire, Emerald.

Each type has 3 identical gems.

You need to arrange 3 Ruby, 3 Sapphire, and 3 Emerald gems in a row. How many different ways can you arrange these 9 gems?

Hint: There are **identical** gems in the set. so think carefully.

Select your answer:

Select your answerOPTIONALTERNATIVE

C(15,3) = 3003

9! = 362880

9!/3!3!3! = 1680

C(9,3) = 84

3! = 6

To determine the number of different ways to arrange 3 Ruby, 3 Sapphire, and 3 Emerald gems in a row, we must consider that there are identical gems in each type. This can be solved using the multinomial coefficient.

The formula for arranging \(n\) items where there are groups of identical items is given by:

\[
\frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}
\]

where \(n\) is the total number of items, and \(n_1, n_2, \ldots, n_k\) are the counts of each identical item.

In this case, we have:

- Total number of gems, \(n = 9\) (3 Rubies + 3 Sapphires + 3 Emeralds)
- Number of identical Rubies, \(n_1 = 3\)
- Number of identical Sapphires, \(n_2 = 3\)
- Number of identical Emeralds, \(n_3 = 3\)

So the number of different ways to arrange these gems is:

\[
\frac{9!}{3! \cdot 3! \cdot 3!}
\]

Calculating this:

\[
9! = 362880
\]
\[
3! = 6
\]
\[
3! \cdot 3! \cdot 3! = 6 \cdot 6 \cdot 6 = 216
\]
\[
\frac{362880}{216} = 1680
\]

So, the correct answer is:
**\(\frac{9!}{3!3!3!} = 1680\)**

Would you like any further details or have any questions?

Here are some additional questions you might want to ask:
1. How do you calculate combinations and permutations?
2. What is the difference between permutations and combinations?
3. Can you provide an example of a problem involving permutations with identical items?
4. How do multinomial coefficients apply to probability?
5. What is the significance of factorials in combinatorics?

**Tip:** When dealing with problems that involve arranging identical items, always use the multinomial coefficient to account for the repetitions.

Learn how to calculate the number of different ways to arrange 3 Ruby, 3 Sapphire, and 3 Emerald gems using combinatorial mathematics. This problem involves factorial calculations and multinomial coefficients. Suitable for advanced high school students.

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