If P, Q, R, S are 4 individuals, how many teams of size exceeding one can be formed, with Q as a member?

The correct option is 7.

Step-by-Step Explanation:

1. Understand the given conditions:

We have a set of 4 individuals: {P, Q, R, S}.

We need to form teams such that:

• The team size must exceed 1 (i.e., the team must have 2, 3, or 4 members).

• Individual Q must be a member of every team.

2. Determine the team formation logic:

Since Q must always be included in the team, the remaining members of any team must be selected from the remaining 3 individuals: {P, R, S}.

3. Calculate the number of valid teams by size:

• Teams of size 2:

Q is already 1 member, so we need to choose 1 more member from the 3 remaining individuals {P, R, S}.

Number of ways =

$C_1^3=3$

The possible teams of size 2 are: {Q, P}, {Q, R}, and {Q, S}.

• Teams of size 3:

Q is already 1 member, so we need to choose 2 more members from the 3 remaining individuals {P, R, S}.

Number of ways =

$C_2^3=3$

The possible teams of size 3 are: {Q, P, R}, {Q, P, S}, and {Q, R, S}.

• Teams of size 4:

Q is already 1 member, so we need to choose all 3 remaining individuals {P, R, S}.

Number of ways =

$C_3^3=1$

The possible team of size 4 is: {Q, P, R, S}.

4. Total Number of Teams:

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

$\text{Total teams}=3+3+1=7$

Thus, a total of 7 teams of size exceeding one can be formed with Q as a member.