Alice, Andrew, and six other students are to be divided into two teams of four people. How many ways are there to form the teams so that Alice and Andrew are on opposite teams

Since Alice, Andrew, and six other students are to be divided into two teams of four people, and we require the number of ways to form the teams so that Alice and Andrew are on opposite teams, this is a combination question since order doesn't matter.

Number of ways of combining the first team

For the first team of 4, if Alice takes the first position, then there are 3 positions left to be occupied by the remaining 3 persons.

So, the number of ways of combining this team is n = 1 × ³C₃

= 1 × 1

= 1 way

Number of ways of combining the second team

Also, for the second team of 4, if Andrew takes the first position, then there are 3 positions left to be occupied by the remaining 3 persons.

So, the number of ways of combining this team is N = 1 × ³C₃

= 1 × 1

= 1 way

Number of ways of combining both teams

So, the number of ways teams can be formed so that Alice and Andrew are on opposite teams is N' = n + N

= 1 + 1

= 2 ways

So, the number of ways there are to form the teams so that Alice and Andrew are on opposite teams is 2 ways.

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