Last Problem of the Week this year! Having a bit of trouble with it: In how many ways can seven basketball players of different heights line up in a single row so that no player is standing between two people taller than she is?
Does it have anything to do with 7! or am I on the wrong track completely?
They could line up in ascending order of height.
Is this the only solution? I do apologise for sounding rude, but they are generally more difficult than they appear.
The other solution is lining up in descending order.
These solutions work. I don't see any reason to complicate this problem. :-)
Check back here to see if another math tutor has other solutions.
Okay. Thank you very much, Ms. Sue, I will.
You're very welcome, Rosa.
To solve this problem, you need to consider the conditions mentioned in the question. Let's break it down step by step:
First, let's determine the number of ways to arrange the players in a single row without any restrictions. Since all seven players have different heights, the number of possible arrangements is simply 7!
Now, let's consider the given condition that no player can be standing between two people taller than them. This means that players can only be arranged in non-decreasing order of height.
To visualize this, let's assign a number to each player representing their height. For example, let's denote the shortest player as 1 and the tallest player as 7.
Now, the question asks us to find the number of ways to arrange the players, so that no player has any taller players behind them.
To solve this, start with the tallest player (7) and place them at any position in the row. Then, place the second tallest player (6) anywhere to the left of the tallest player.
Continuing this process, you would place the third tallest player (5) anywhere to the left of the second tallest player, and so on, until you place the shortest player (1) anywhere to the left of the second-to-last player.
The number of ways to arrange the players following these conditions can be represented by the multiplication of the available positions for each player. It can be calculated as follows:
Number of ways = 1 * 2 * 3 * 4 * 5 * 6 * 7
This simplifies to 7!.
Therefore, you were correct in your initial intuition! The number of ways to arrange the seven basketball players, such that no player is standing between two people taller than themselves, is indeed 7!.