Two elementary school teachers have 24 stu-dents  each.  The  first  teacher  splits  his  students  into  four 

groups of six. The second teacher splits her students into six 

groups of four. Which teacher has more options?

To determine which teacher has more options for splitting their students into groups, we need to calculate the number of ways each teacher can arrange their students into the specified groups. 

### Step 1: Calculate the number of ways for the first teacher

The first teacher splits 24 students into 4 groups of 6 students each. The number of ways to do this is calculated using the multinomial coefficient:

\[
\text{Number of ways} = \frac{24!}{(6!)^4 \cdot 4!}
\]

Here:
- \(24!\) represents the total number of ways to arrange 24 students.
- \((6!)^4\) accounts for the fact that each group of 6 can be arranged internally in \(6!\) ways, and there are 4 such groups.
- \(4!\) accounts for the fact that the order of the groups themselves does not matter.

### Step 2: Calculate the number of ways for the second teacher

The second teacher splits 24 students into 6 groups of 4 students each. The number of ways to do this is:

\[
\text{Number of ways} = \frac{24!}{(4!)^6 \cdot 6!}
\]

Here:
- \(24!\) represents the total number of ways to arrange 24 students.
- \((4!)^6\) accounts for the fact that each group of 4 can be arranged internally in \(4!\) ways, and there are 6 such groups.
- \(6!\) accounts for the fact that the order of the groups themselves does not matter.

### Step 3: Compare the two quantities

While the formulas might seem complex, the comparison between the two can be made by analyzing their components. Since \(6!\) is larger than \(4!\) and there are more groups (6 groups of 4 compared to 4 groups of 6), the multinomial coefficient for the second teacher will generally be smaller. The reason is that as the number of groups increases and the size of each group decreases, the number of possible groupings diminishes.

Therefore, the first teacher, who groups students into 4 groups of 6, will have **more options**.

### Final Answer:
The **first teacher** has more options for splitting their students into groups.

Would you like more details or have any questions?

Here are some related questions to consider:
1. How would the number of options change if the first teacher grouped students into 3 groups of 8?
2. If both teachers had 30 students instead of 24, how would the number of options compare?
3. What if the first teacher groups into 2 groups of 12, and the second into 12 groups of 2?
4. How does the total number of students affect the number of grouping options?
5. What if the second teacher had 28 students and split them into 7 groups of 4?

**Tip:** The number of grouping options is greatly affected by both the size of the groups and the number of groups. Multinomial coefficients are key to solving these problems.

Explore the comparison of grouping options in elementary school mathematics. Analyze scenarios where teachers split students into different groups based on given criteria, calculating the number of possible arrangements using combinatorial mathematics. Suitable for Grades 6-8.

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