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Coach Miller is filling out the starting lineup for his [#permalink]

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17 Jan 2005, 14:40

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Coach Miller is filling out the starting lineup for his indoor soccer team. There are 10 boys on the team, and he must assign 6 starters to the following positions: 1 goalkeeper, 2 on defense, 2 in midfield, and 1 forward. Only 2 of the boys can play goalkeeper, and they cannot play any other positions. The other boys can each play any of the other positions. How many different groupings are possible?

It is not clear to me how order of picking 2 on defense, 2 in midfield etc matters, for me it was just number of ways for creating a team by picking 1 goalkeeper out of 2 i.e. 2C1 way and rest of 5 members from 8 ppl, in 8C5. Total of 112 ways

2c1*8c2*6c2*4c1
but taking like this my question is how are we distinguishing between the goal keepers and the rest in the sense.. the 2 eligible goal keepers cannot play the other positions where as allt he other can..

When you consider the first choice 2C1 to select the goalkeepers then you put then out of the pool in filling the next position in 8C2 ways. We do not care wo they really are, just that 2 of them are allocated to this job and then are not part of the remaining players choice.

It is not clear to me how order of picking 2 on defense, 2 in midfield etc matters, for me it was just number of ways for creating a team by picking 1 goalkeeper out of 2 i.e. 2C1 way and rest of 5 members from 8 ppl, in 8C5. Total of 112 ways

banerjeea_98, you are OK so far, but you have not finished: you need to arrange those 5 you pick for the team in their categories: 5C3 * 3C2 *1C1.

So total 2C1 * 8C5 * 5C2 * 3C2 1*1 = 112 * 30 = 3360

When you consider the first choice 2C1 to select the goalkeepers then you put then out of the pool in filling the next position in 8C2 ways. We do not care wo they really are, just that 2 of them are allocated to this job and then are not part of the remaining players choice.