A baseball coach has a team of 22 players .In how many ways can be select a lineup of 9 players to start the game. ( order matters)
22P9 ... 22! / [(22 - 9)! ]
To calculate the number of ways to select a lineup of 9 players from a team of 22 players (where order matters), we can use the concept of permutations.
A permutation is an arrangement of objects in a specific order. In this case, we want to arrange 9 players from a total of 22 players.
The formula to calculate the number of permutations is:
nPk = n! / (n - k)!
Where:
- n is the total number of players (22 in this case)
- k is the number of players to be selected (9 in this case)
- "!" denotes the factorial function, which means multiplying a number by all positive integers less than itself down to 1.
So, plugging in the values into the formula, we have:
22P9 = 22! / (22 - 9)!
Now, let's calculate step by step:
First, calculate the factorial of 22:
22! = 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Next, calculate the factorial of (22 - 9) = 13:
13! = 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Now, divide the factorial of 22 by the factorial of 13:
22P9 = 22! / 13!
Simplifying the expression:
22P9 = (22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13!) / 13!
The 13! terms cancel out:
22P9 = 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14
So, the number of ways to select a lineup of 9 players from a team of 22 players is 22,230,876,320.