How many ways can seven basketball players of different heights line up in a single row so no player is standing between two players taller then herself?
To find the number of ways the seven basketball players can line up in a single row such that no player is standing between two players taller than herself, we can use combinatorics.
Let's break down the problem step by step:
Step 1: Determine the tallest player
In this case, we have seven basketball players of different heights. We need to find the tallest player. Let's assume Player 7 is the tallest.
Step 2: Determine the position of the tallest player
Since no player can stand between two players taller than herself, the tallest player must be at either end of the row. So, the tallest player can either be in the first position or the seventh position.
Step 3: Arrange the remaining players
We have six remaining players (Player 1 to Player 6) to arrange in the row. Since no player can stand between two taller players, the shorter players must be placed in a decreasing order of height. One approach is to start from one end of the row and place the shorter players in descending order of height towards the other end.
Step 4: Calculate the number of arrangements
Since the tallest player can be placed in either the first or the seventh position, there are two possibilities. For each possibility, we have 6! (6 factorial) ways to arrange the remaining players.
So, the total number of ways the seven basketball players can line up in a single row, such that no player is standing between two players taller than herself, is:
2 * 6! = 2 * 6 * 5 * 4 * 3 * 2 * 1 = 144
Therefore, there are 144 different ways the basketball players can line up in this scenario.