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 basketball players can line up, satisfying the given condition, we can use the principle of permutation.
Step 1: Determine the tallest player's position:
The tallest player must stand at either end of the row since no one can be taller than them. Therefore, we have 2 options for the position of the tallest player.
Step 2: Determine the position of the second tallest player:
Since no one can be taller than the second tallest player, they must stand next to the tallest player. Since the tallest player is already fixed at one end, we have one option for the position of the second tallest player.
Step 3: Determine the remaining positions:
After placing the tallest and second tallest players, we are left with 5 players and 5 positions remaining. We can arrange these 5 players among the remaining 5 positions in any order. This can be done in 5! (factorial) ways.
Step 4: Multiply the possibilities:
To get the total number of ways, we multiply the number of options for each step.
Total ways = Step 1 × Step 2 × Step 3
= 2 × 1 × 5!
= 2 × 1 × 120
= 240
Therefore, there are 240 different ways the seven basketball players can line up in a single row, satisfying the given condition.
To solve this problem, we can use the concept of permutations. Let's break down the problem step by step:
Step 1: Identify the total number of players.
In this case, we have seven basketball players.
Step 2: Identify the players who cannot stand between two taller players.
According to the problem, no player should stand between two players taller than themselves. Let's call these players the "protected" players.
In this case, the protected players are the tallest player and the shortest player. So we have:
- Tallest player: Cannot have a taller player standing to their right.
- Shortest player: Cannot have a taller player standing to their left.
Step 3: Calculate the number of ways to arrange the protected players.
Since the tallest player cannot have a taller player to their right and the shortest player cannot have a taller player to their left, they are fixed in their respective positions.
So, the number of ways to arrange the protected players is 2. (Tallest player on the left, shortest player on the right OR shortest player on the left, tallest player on the right)
Step 4: Arrange the remaining players.
After fixing the positions of the protected players, we are left with five players. Since there are no restrictions on how these players can be arranged, we can arrange them in any order.
The number of ways to arrange the remaining five players is 5!.
Step 5: Calculate the final result.
To get the total number of possible arrangements, we need to multiply the results from steps 3 and 4.
Total number of arrangements = Number of arrangements for protected players × Number of arrangements for remaining players
Total number of arrangements = 2 × 5! = 2 × 5 × 4 × 3 × 2 × 1 = 240
So, there are 240 different ways the seven basketball players can line up in a single row, satisfying the given condition.
Two would be ascending or descending order.
Twelve more would be ascending or descending order, with any one of the not-tallest going to the other end.
I can think of four others that involve flipping and inverting more than one to the opposite side. Maybe you can think of others. Add them up
By seeing how many arrangements I could make with N players, I found that for N = 1, 2, 3, 4, and 5 players, the number of arrangements is 2^(N-1).
If this trend continues, the number of possible arrangements of N = 7 players is 2^6 = 64. Figuring it out with numbers (1 through 7) is easier than doing it with "straws" of different length.