Final answer:
There are 140 ways to select a team of four members with at most one boy and a captain from a group consisting of six girls and four boys.
Step-by-step explanation:
To select a team of four members from a group consisting of six girls and four boys, including the selection of a captain among the four members, while ensuring that the team has at most one boy, we need to consider two cases:
Case 1: The team consists of four girls.
The number of ways to select four girls from the six available is given by the combination formula: C(6, 4) = 6! / (4!(6-4)!) = 15.
In this case, the captain can be selected from the four girls, so the number of ways to select a captain is 4.
So, the total number of ways to select a team with four girls and a captain from the girls is 15 * 4 = 60.
Case 2: The team consists of three girls and one boy.
The number of ways to select three girls from the six available is given by the combination formula: C(6, 3) = 6! / (3!(6-3)!) = 20.
The number of ways to select one boy from the four available is given by the combination formula: C(4, 1) = 4! / (1!(4-1)!) = 4.
In this case, the captain can be selected from the four members of the team, which includes the three girls and the one boy, so the number of ways to select a captain is 4.
So, the total number of ways to select a team with three girls, one boy, and a captain is 20 * 4 = 80.
Therefore, the total number of ways to select the team, considering the two cases, is 60 + 80 = 140.