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The number of six-student teams possible where a team consists of three freshmen and three sophomores, from a group of 12 students consisting of five freshmen and seven sophomores using combinations is computed to be 350.
The combination is a process of calculating the number of ways of selecting a smaller set, from a larger set, when the order of selection is irrelevant.
In selecting x number of items, from n number of items, when the order of selection is irrelevant, we use the combination, to calculate the number of possible ways as follows:
nCx = n!/{(x!)((n - x)!)}.
In the question, we are asked for the number of six-student teams possible where a team consists of three freshmen and three sophomores, from a group of 12 students consisting of five freshmen and seven sophomores.
The number of ways of selecting 3 freshmen from 5, using a combination is:
5C3 = 5!/{(3!)((5 - 3)!} = 120/{6*2} = 120/12 = 10.
The number of ways of selecting 3 sophomores from 7, using a combination is:
7C3 = 7!/{(3!)((7 - 3)!} = 5040/{6*24} = 5040/144 = 35.
The total number of teams possible is the product of each.
Thus, the total number of teams possible is 5C3 * 7C3 = 10*35 = 350.
Thus, the number of six-student teams possible where a team consists of three freshmen and three sophomores, from a group of 12 students consisting of five freshmen and seven sophomores using combinations is computed to be 350.
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