There are 10 girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?

pls explain it to a 13 years old kid

Sure! Let's break it down step by step in a simpler way.

### Problem Understanding

You have 10 girls, and each girl is a different height. You want to line them up in such a way that no girl is standing directly between two girls who are shorter than her. This means, in any group of three girls standing next to each other, the one in the middle can't be taller than the other two.

### Key Idea: The Zig-Zag Pattern

To solve this, think about a "zig-zag" or "wave" pattern. If a line goes up and down like a wave, that's how we want the heights of the girls to be arranged.

For example:
- If a girl is taller than the one before her, the next girl should be shorter.
- If a girl is shorter than the one before her, the next girl should be taller.

This creates a pattern where the heights go up and down, like climbing up a hill and then going down.

### Counting the Patterns

To figure out how many ways there are to arrange the girls like this, mathematicians have figured out that this problem is related to something called **alternating permutations**. These are special arrangements where the heights alternate between higher and lower.

For 10 girls, the number of possible zig-zag patterns (or alternating permutations) has already been calculated, and the answer is **768**.

### Summary

So, there are **768 different ways** to arrange the 10 girls in a line so that no girl is standing between two shorter girls.

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Do you have any questions or need more details?

Here are 5 related questions you might find interesting:
1. What are other examples of zig-zag patterns in everyday life?
2. How would this problem change if we had only 3 or 4 girls?
3. Can you think of a way to write down all possible arrangements for a small number of girls?
4. What if you want the pattern to be only increasing or only decreasing? How many ways can you arrange them?
5. Why do you think this kind of pattern is called an "alternating permutation"?

**Tip:** When solving puzzles or problems, look for patterns that repeat themselves—this often leads you to the solution!

Discover how to arrange 10 girls in a queue so that no girl stands between two shorter girls. Learn about alternating permutations and the zig-zag pattern in combinatorics. Suitable for Grades 7-9.

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