# How many possible configurations of a soccer team is there given that each of the three outfield position needs at least 2 players?

I have been struggling with this question for over a day. I think I am on the correct track, but I am not certain.

My approach was to use stars and bars for the representation of the team. I places the first bar so that the total to the left was exactly 2 and counted the spaces that the second bar could go while leaving the right most segment with at least 2 players.

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The total ended up being 5, so I have P(5,1) places for the bar to go. I repeated this process moving the initial bar one space to the right each time and summed the results, which I found to be P(5,1)+P(4,1)+P(3,1)+P(2,1)=5+4+3+2=14.

Am I correct or have I made a mistake? Thank you for your help. This is my first semester of combinatorics and the first time I have done math like this.

$$14$$ is the wrong answer. We have two players reserved for each of the defender, midfielder and forward rows, so we might as well ignore them. We have $$4$$ players left to distribute however we like among the three positions – this is where we apply stars and bars: 4 stars, 2 bars, $$\binom{4+2}2=\binom62=15$$.
• @Goldsten More properly, the calculation is $\binom{4+3-1}4$, there being $4$ leftover players and $3$ positions. Jan 9, 2020 at 16:57