 Maxwell thinks of a way to make a special tea. In order
to make this drink, Maxwell must perform the steps in the following action: Maxwell will crush
the red flower into a bottle, then Maxwell will pull the green flower’s petals and place them
into the bottle, and finally Maxwell will place the entire blue flower on top of the other flowers.
Maxwell has also placed the flowers corresponding with each drink next to its respective bottle.
In other words, the flowers are labelled from 1 − 36.
For each of the 36 herbal drinks, Maxwell must perform these three steps in this exact chronological order or else the drink will mix improperly and lose its taste. However, Maxwell may
choose to work on and/or finish another drink before finishing the drink Maxwell is currently
on. In how may distinct orders can Maxwell create all 36 special teas?
For example, to create herbal drink 1 and herbal drink 2, Maxwell could process:
(a) Red flower for drink 1
(b) Red flower for drink 2
(c) Green flower for drink 2
(d) Blue flower for drink 2
(e) Green flower for drink 1
(f) Blue flower for drink 1

Maxwell's task can be broken down into a combinatorics problem where each of the 36 drinks follows three specific steps in a fixed order. The main challenge is determining how many distinct orders Maxwell can use to process all 36 drinks, given the sequence constraints for each drink.

### Problem Analysis

For each drink, Maxwell must follow three specific steps:

1. Crush the red flower.
2. Pull the green flower’s petals and place them into the bottle.
3. Place the blue flower on top of the other flowers.

The order of these steps for each drink must be maintained, but Maxwell is allowed to switch between different drinks at any point in time. For example, Maxwell could do step 1 for drink 1, then step 1 for drink 2, then step 2 for drink 1, and so on, as long as the sequence of steps within each individual drink is respected.

### Total Number of Steps

Each drink requires 3 steps, and there are 36 drinks, so Maxwell needs to perform a total of:
\[
3 \times 36 = 108 \text{ steps}.
\]
Out of these 108 steps, we need to determine how Maxwell can sequence them while maintaining the required order for each drink.

### Distinct Orders

For each drink, the three steps must always occur in the order "red → green → blue." Therefore, we can think of this as partitioning the 108 steps into three groups for each drink, where the steps within each group must be in a specific order.

Thus, the problem reduces to finding the number of ways to permute 108 steps, where:
- 36 steps are for the red flowers (one for each drink),
- 36 steps are for the green flowers (one for each drink), and
- 36 steps are for the blue flowers (one for each drink).

Since these steps are indistinguishable except for their relative order within each group, the number of distinct sequences can be computed using multinomial coefficients. Specifically, the number of ways to arrange the 108 steps is given by:
\[
\frac{108!}{36! \times 36! \times 36!}.
\]

### Final Answer

The total number of distinct ways Maxwell can perform the steps for all 36 drinks is:
\[
\boxed{\frac{108!}{36! \times 36! \times 36!}}.
\]

This is the final expression for the number of distinct orders in which Maxwell can create all 36 special teas.

### Next Steps and Exploration

Do you want to go into more details on the calculation or explore related concepts?

Here are 5 related questions to expand your understanding:
1. How would the solution change if Maxwell could perform the steps in any order for each drink?
2. What if Maxwell had only 10 drinks to prepare—how would the result be different?
3. How can multinomial coefficients be used in other types of ordering or arrangement problems?
4. Can you calculate an approximation for \(\frac{108!}{36! \times 36! \times 36!}\) using Stirling's approximation?
5. How would the problem change if each flower type had a different number of steps?

**Tip:** When dealing with large factorials, approximations like Stirling’s approximation can help simplify calculations.

Maxwell thinks of a way to make a special tea. In order to make this drink, Maxwell must perform the steps in the following action: Maxwell will crush the red flower into a bottle, then Maxwell will pull the green flower’s petals and place them into the bottle, and finally Maxwell will place the entire blue flower on top of the other flowers. Maxwell has also placed the flowers corresponding with each drink next to its respective bottle. In other words, the flowers are labelled from 1 − 36. For each of the 36 herbal drinks, Maxwell must perform these three steps in this exact chronological order or else the drink will mix improperly and lose its taste. However, Maxwell may choose to work on and/or finish another drink before finishing the drink Maxwell is currently on. In how may distinct orders can Maxwell create all 36 special teas?

Solve the combinatorics problem involving Maxwell's special tea preparation. Maxwell must follow specific steps to create 36 herbal drinks in distinct orders. This problem explores the use of multinomial coefficients and factorials to find the number of possible sequences. Perfect for college-level students studying combinatorics.

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