Calculating Teams Including Q

The problem asks for the number of teams that can be formed from four individuals (P, Q, R, S) such that:

• The team size is greater than 1 (i.e., 2, 3, or 4 members).
• Q is always a member of the team.

Since Q must be in the team, we need to select the remaining members from the other individuals (P, R, S).

Teams of Size 2 with Q

A team of size 2 requires Q plus one more member.

We need to choose 1 additional member from the remaining 3 individuals (P, R, S).

The number of ways to do this is calculated using combinations: $\binom{3}{1}$.

Calculation: $\binom{3}{1} = \frac{3!}{1!(3-1)!} = 3$.

Teams of Size 3 with Q

A team of size 3 requires Q plus two other members.

We need to choose 2 additional members from the remaining 3 individuals (P, R, S).

The number of ways is calculated using combinations: $\binom{3}{2}$.

Calculation: $\binom{3}{2} = \frac{3!}{2!(3-2)!} = 3$.

Teams of Size 4 with Q

A team of size 4 requires Q plus all three remaining members.

We need to choose 3 additional members from the remaining 3 individuals (P, R, S).

The number of ways is calculated using combinations: $\binom{3}{3}$.

Calculation: $\binom{3}{3} = \frac{3!}{3!(3-3)!} = 1$.

Total Number of Teams

To find the total number of possible teams, we sum the possibilities for each team size:

Total Teams = (Teams of size 2) + (Teams of size 3) + (Teams of size 4)

Total Teams = 3 + 3 + 1 = 7.

Therefore, there are 7 possible teams of size exceeding one that include Q.