Comprehension
Members: \( K, L, M, N, P, Q, R, S, U, W \) are the only ten members in a department.
There is a proposal to form a team from within the members of the department, subject to the following conditions:
- A team must include exactly one among \( P, R, S \).
- A team must include either \( M \) or \( Q \), but not both.
- If a team includes \( K \), then it must also include \( L \), and vice versa. \[ K \in \text{Team} \iff L \in \text{Team} \]
- If a team includes one among \( S, U, W \), then it must also include the other two. \[ (S \in T) \lor (U \in T) \lor (W \in T) \Rightarrow S, U, W \in T \]
- \( L \) and \( N \) cannot be members of the same team.
- \( L \) and \( U \) cannot be members of the same team.
The size of a team is defined as the number of members in the team.
Question: 1
Who cannot be a member of a team of size 3?
Who cannot be a member of a team of size 3?
Show Hint
When team formation constraints are given, check for “chain reactions” where including one member forces inclusion of others, which might violate size limits.
Updated On: Jul 31, 2025
- L
- M
- N
- P
- Q
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The Correct Option is A
Solution and Explanation
From the conditions:
- If K is included, then L must be included, and vice versa.
- If one among S, U, and W is included, then all three must be included.
- L and N cannot be in the same team.
For a team of size 3, including L would force the inclusion of K (Condition 3), making at least 2 members. Additionally, one of P, R, or S must be included (Condition 1), but if S is included, then U and W must also be included (Condition 4), increasing size beyond 3. Thus, L cannot be part of a 3-member team.
- If one among S, U, and W is included, then all three must be included.
- L and N cannot be in the same team.
For a team of size 3, including L would force the inclusion of K (Condition 3), making at least 2 members. Additionally, one of P, R, or S must be included (Condition 1), but if S is included, then U and W must also be included (Condition 4), increasing size beyond 3. Thus, L cannot be part of a 3-member team.
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Question: 2
Who can be a member of a team of size 5?
Who can be a member of a team of size 5?
Show Hint
Always check if the team size can accommodate all compulsory inclusions when a member is chosen.
Updated On: Jul 31, 2025
- K
- L
- M
- P
- R
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The Correct Option is B
Solution and Explanation
In a team of size 5, L can be included if K is also included (Condition 3). The other positions can be filled without violating Conditions 1, 2, 4, 5, and 6. The size is sufficient to accommodate all forced inclusions when L is selected, so L is possible.
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Question: 3
What would be the size of the largest possible team?
What would be the size of the largest possible team?
Show Hint
When maximizing team size, look for selections that trigger the largest forced inclusions without violating constraints.
Updated On: Jul 31, 2025
- 8
- 7
- 6
- 5
- Cannot be determined
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The Correct Option is B
Solution and Explanation
The maximum team would include:
- Exactly one among P, R, S (Condition 1).
- Either M or Q (Condition 2).
- If K is included, then L must also be included (Condition 3).
- If S is included, U and W must also be included (Condition 4).
- L and N cannot be together (Conditions 5 and 6).
Constructing the largest team: choose S, U, W together (3 members), include K and L (2 more), add M (1 more), and N (1 more). This totals \(3 + 2 + 1 + 1 = 7\) members.
- Either M or Q (Condition 2).
- If K is included, then L must also be included (Condition 3).
- If S is included, U and W must also be included (Condition 4).
- L and N cannot be together (Conditions 5 and 6).
Constructing the largest team: choose S, U, W together (3 members), include K and L (2 more), add M (1 more), and N (1 more). This totals \(3 + 2 + 1 + 1 = 7\) members.
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Question: 4
What could be the size of a team that includes K?
What could be the size of a team that includes K?
Show Hint
Check all possible combinations when a member triggers other inclusions — sometimes only one exact team size will work.
Updated On: Jul 31, 2025
- 2 or 3
- 2 or 4
- 3 or 4
- Only 2
- Only 4
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The Correct Option is
Solution and Explanation
If K is included, then L must also be included (Condition 3). With L present, N is excluded (Condition 5). One among P, R, S must be chosen; choosing S would require U and W (Condition 4), making size>4. Thus, avoiding S and choosing either P or R, along with M or Q, leads to exactly 4 members.
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Question: 5
In how many ways can a team be constituted so that the team includes N?
In how many ways can a team be constituted so that the team includes N?
Show Hint
When counting valid teams, fix the required member first, then apply constraints to eliminate invalid combinations.
Updated On: Jul 31, 2025
- 2
- 3
- 4
- 5
- 6
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The Correct Option is B
Solution and Explanation
If N is included, L is excluded (Condition 6). This allows flexibility with K (since L is not needed), and one among P, R, S must be selected. If S is chosen, U and W must also be chosen (Condition 4). Counting valid combinations with other members respecting M/Q choice gives exactly 3 possible teams.
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