Tina menghias kamarnya menggunakan balon berwarna-wárni. Tina mempunyai 5 balon merah, 3. balon b...

Introduction

In this tutorial, we will explore how Tina decorates her room using colorful balloons while solving a mathematical problem related to permutations. This tutorial is particularly relevant for students studying probability and permutations in mathematics, especially at the high school level.

Step 1: Understand the Problem

Tina has the following balloons:

• 5 red balloons
• 3 blue balloons
• 4 yellow balloons

The goal is to arrange these balloons in a line with the following conditions:

• The first and last balloons must be yellow.
• The blue balloons must be adjacent to each other.

Practical Tip

Before proceeding, write down the total number of balloons and review the positioning requirements.

Step 2: Visualize the Arrangement

To simplify the arrangement, let's denote the balloons:

• R (Red)
• B (Blue)
• Y (Yellow)

Since the yellow balloons must occupy the first and last positions, we can represent the arrangement as:

• Y _ _ _ _ _ _ _ _ _ Y

This layout implies that we need to arrange the remaining balloons (5 red, 3 blue) in the 9 spaces between the two yellow balloons.

Step 3: Treat Blue Balloons as a Single Unit

Since the blue balloons need to be adjacent, we can treat them as a single unit. This means we can group the 3 blue balloons together as one unit (BBB).

Now, our arrangement looks like this:

• Y [BBB] R R R R R Y

Now we can count the effective units:

• 1 (BBB) + 5 (R) = 6 units

Step 4: Calculate the Total Arrangements

We now need to calculate how many different ways we can arrange these units. The total number of arrangements can be calculated using the formula for permutations of multiset:

[ P = \frac{n!}{n_1! \cdot n_2! \cdot ...} ]

Where:

• n = total number of units (6)
• n1 = number of identical units (5 red balloons)

Calculation

[ P = \frac{6!}{5! \cdot 1!} = \frac{720}{120} = 6 ]

Step 5: Consider the Blue Balloon Arrangements

Since the blue balloons can be arranged among themselves, we also need to account for the arrangements of the blue balloons (3!):

[ 3! = 6 ]

Step 6: Final Calculation

Finally, the total arrangements will be the product of the arrangements of the units and the arrangements of the blue balloons:

[ Total Arrangements = 6 \cdot 6 = 36 ]

Conclusion

In this tutorial, we learned how to arrange balloons while adhering to specific conditions using permutations. The final answer is that there are 36 unique ways to arrange Tina's balloons with the given constraints. This process can be applied to similar problems in probability and combinatorial mathematics. For further practice, try creating your own arrangements with different quantities or colors of balloons.