A certain professional soccer team runs a 4-4-2 formation on the field. This means they play with 4 defenders, 4 midfielders, 2 forwards, and 1 goalkeeper on the field. If the team has 6 defenders, 5 midfielders, 3 forwards, and 2 goalkeepers, how many different groups of 11 starting players could they have?
note that:Each player only plays in his designated position. The 2nd goalkeeper is not allowed to play as a defender.
just find out how many possibilities there are for each position, and multiply them all together:
C(6,4) * C(5,4) * C(3,2) * C(2,1) = 450
If your comment about the 2nd goalkeeper means that he may play as a midfielder or a forward, then things get more complicated.
ur answer is totally correct....thank u very much
To find the number of different groups of 11 starting players, we need to consider a few conditions.
First, we need to ensure that the team maintains the 4-4-2 formation. This means there should be exactly 4 defenders, 4 midfielders, and 2 forwards in the starting lineup. Additionally, we need to ensure that only one goalkeeper is included in the starting lineup.
Let's break this down step by step:
1. Choose 4 defenders out of the 6 available:
The number of ways to choose 4 defenders from a group of 6 can be calculated using combinations. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to be chosen.
In this case, we have 6 defenders and we want to select 4. So the number of options for choosing 4 defenders would be:
6C4 = 6! / (4!(6-4)!) = 15 ways
2. Choose 4 midfielders out of the 5 available:
Similarly, we have 5 midfielders and we want to select 4. So the number of options for choosing 4 midfielders would be:
5C4 = 5! / (4!(5-4)!) = 5 ways
3. Choose 2 forwards out of the 3 available:
We have 3 forwards and we want to select 2. So the number of options for choosing 2 forwards would be:
3C2 = 3! / (2!(3-2)!) = 3 ways
4. Choose 1 goalkeeper out of the 2 available:
Since we only want one goalkeeper in the starting lineup, we don't need to calculate any combinations for this step. There are simply 2 options for choosing the goalkeeper.
5. Multiply the number of options for each position together:
To get the total number of different groups of 11 starting players, we need to multiply the number of options for each position together.
Total groups = (Number of defender options) * (Number of midfielder options) * (Number of forward options) * (Number of goalkeeper options)
Total groups = 15 * 5 * 3 * 2 = 450
So, the team could have 450 different groups of 11 starting players that satisfy the given conditions.