My soccer team has 16 players. I have to choose a starting lineup of a goalie and 10 regular players (the regular players are interchangeable). How many different starting lineups can I choose?
To find the number of different starting lineups you can choose, we need to use the concept of combinations. In this case, we are selecting 10 regular players out of 16, which can be represented as "16 choose 10", or written as 16C10.
To calculate this combination, we can use the formula:
nCr = n! / [(n - r)! * r!]
where n is the total number of players available (16 in this case), and r is the number of players to be selected (10 in this case). The exclamation mark (!) denotes the factorial operation.
Using this formula, we can calculate the number of different starting lineups as follows:
16C10 = 16! / [(16 - 10)! * 10!]
= 16! / [6! * 10!]
Now, let's break down the factorial expressions:
16! = 16 * 15 * 14 * 13 * 12 * 11 * 10!
6! = 6 * 5 * 4 * 3 * 2 * 1
By simplifying the expression, we get:
16C10 = (16 * 15 * 14 * 13 * 12 * 11 * 10!) / [(6 * 5 * 4 * 3 * 2 * 1) * 10!]
= (16 * 15 * 14 * 13 * 12 * 11) / (6 * 5 * 4 * 3 * 2 * 1)
= 32,432
Therefore, you can choose from 32,432 different starting lineups.
To determine the number of different starting lineups, we need to calculate the combinations of players that can be chosen for each position.
First, let's choose the goalkeeper, which is only one position. You have 16 players to choose from initially, but once you choose a goalkeeper, there will be 15 players left to choose from.
Next, let's choose the 10 regular players from the remaining 15 players. For this, we need to calculate the number of combinations. The formula for calculating combinations is nCr, where n is the number of players to choose from, and r is the number of players to be chosen.
Using the combination formula, we can calculate the number of combinations as follows:
10 regular players: 15C10 = 3003
Finally, to calculate the total number of different starting lineups, we multiply the number of goalkeepers by the number of combinations of regular players:
Total number of lineups = 1 (goalkeeper) * 3003 (regular players)
Therefore, you can choose a total of 3003 different starting lineups.