Alice and her three best friends are having a picnic. Alice brought 12 multicolor carrots to share.
How many ways are there to divide the carrots evenly among Alice and her three best friends?

To solve this problem, we need to divide 12 multicolor carrots evenly among Alice and her three best friends. Since there are 4 people (Alice and her three friends), we need to divide the 12 carrots such that each person gets an equal share.

### Step-by-step solution:

1. **Total number of carrots:** 12
2. **Number of people:** 4 (Alice + 3 friends)
3. **Each person's share:** \( \frac{12}{4} = 3 \) carrots per person.

Since the carrots are multicolored, they are distinct (they could differ in color). Therefore, the number of ways to distribute 12 distinct carrots into four groups of 3 carrots each is the same as the number of ways to partition 12 objects into four groups of size 3.

### Formula:
This is a combinatorial problem involving partitions of distinct objects. The number of ways to divide 12 distinct objects into groups of size 3 is given by the multinomial coefficient:

\[
\binom{12}{3,3,3,3} = \frac{12!}{3! \cdot 3! \cdot 3! \cdot 3!}
\]

Now let's compute this:

\[
12! = 479,001,600
\]
\[
3! = 6
\]
So,
\[
\frac{12!}{(3!)^4} = \frac{479,001,600}{6^4} = \frac{479,001,600}{1,296} = 369,600
\]

Thus, the number of ways to divide the 12 carrots evenly among Alice and her three best friends is **369,600**.

---

Would you like further clarification on any step or more details?

Here are some related questions to consider:
1. What is the general formula for dividing distinct objects into groups of equal sizes?
2. How would the result change if the carrots were identical?
3. What if there were more than 4 friends? How would the solution change?
4. Can we generalize the solution for any number of objects and groups?
5. How does this problem differ if we are dealing with an unequal number of objects?

**Tip:** Always verify whether the objects you are dealing with are identical or distinct, as this impacts the counting methods used.

Alice and her three best friends are having a picnic. Alice brought 12 multicolor carrots to share. How many ways are there to divide the carrots evenly among Alice and her three best friends?

Explore how to solve a combinatorial problem of dividing 12 multicolor carrots evenly among 4 people. Using the multinomial coefficient formula, this step-by-step solution calculates the number of ways to distribute the distinct carrots. Suitable for students in Grades 10-12.

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