A company must form a five-member project team from nine employees: four engineers (A, B, C and D), three designers (E, F and G) and two analysts (H and I). The team must satisfy all of the following conditions:
1. It must include at least two engineers, at least one designer and at least one analyst.
2. A and E cannot both be in the team.
3. A and D cannot both be in the team.
4. F and I cannot both be in the team.
5. If B is in the team, then H must also be in the team.
6. C is in the team if and only if G is in the team.
1. It must include at least two engineers, at least one designer and at least one analyst.
2. A and E cannot both be in the team.
3. A and D cannot both be in the team.
4. F and I cannot both be in the team.
5. If B is in the team, then H must also be in the team.
6. C is in the team if and only if G is in the team.
Question 1
Numerical answer
How many different teams can be formed?
Show answer and solution
Answer: 13
Listing the valid teams. Split by whether C and G (who come as a pair) are included.
*With C and G:* three more members, with at least one more engineer and at least one analyst. Checking all cases gives 11 teams: ABCGH, ACFGH, ACGHI, BCDGH, BCEGH, BCFGH, BCGHI, CDEGH, CDEGI, CDFGH, CDGHI.
*Without C and G:* engineers come from A, B, D (A and D not together) and designers from E, F. With A: the other engineer must be B, so H is in; the designer cannot be E, so F, and then the fifth member... A, B, F, H plus one of D/E/I — all forbidden (A–D, A–E, F–I) ✗. Without A: B and D, so H is in; the fifth is from E, F, I: BDEFH and BDEHI (BDFHI violates F–I). That gives 2 teams.
Total = 11 + 2 = 13 teams.
Answer: 13.
*With C and G:* three more members, with at least one more engineer and at least one analyst. Checking all cases gives 11 teams: ABCGH, ACFGH, ACGHI, BCDGH, BCEGH, BCFGH, BCGHI, CDEGH, CDEGI, CDFGH, CDGHI.
*Without C and G:* engineers come from A, B, D (A and D not together) and designers from E, F. With A: the other engineer must be B, so H is in; the designer cannot be E, so F, and then the fifth member... A, B, F, H plus one of D/E/I — all forbidden (A–D, A–E, F–I) ✗. Without A: B and D, so H is in; the fifth is from E, F, I: BDEFH and BDEHI (BDFHI violates F–I). That gives 2 teams.
Total = 11 + 2 = 13 teams.
Answer: 13.
Question 2
Single correct option
If I is in the team, how many different teams are possible?
- A. 3
- B. 4
- C. 5
- D. 6
Show answer and solution
Answer: (C) 5
Listing the valid teams. Split by whether C and G (who come as a pair) are included.
*With C and G:* three more members, with at least one more engineer and at least one analyst. Checking all cases gives 11 teams: ABCGH, ACFGH, ACGHI, BCDGH, BCEGH, BCFGH, BCGHI, CDEGH, CDEGI, CDFGH, CDGHI.
*Without C and G:* engineers come from A, B, D (A and D not together) and designers from E, F. With A: the other engineer must be B, so H is in; the designer cannot be E, so F, and then the fifth member... A, B, F, H plus one of D/E/I — all forbidden (A–D, A–E, F–I) ✗. Without A: B and D, so H is in; the fifth is from E, F, I: BDEFH and BDEHI (BDFHI violates F–I). That gives 2 teams.
Total = 11 + 2 = 13 teams.
Teams containing I: ACGHI, BCGHI, CDEGI, CDGHI, BDEHI → 5. (F can never appear with I.)
*With C and G:* three more members, with at least one more engineer and at least one analyst. Checking all cases gives 11 teams: ABCGH, ACFGH, ACGHI, BCDGH, BCEGH, BCFGH, BCGHI, CDEGH, CDEGI, CDFGH, CDGHI.
*Without C and G:* engineers come from A, B, D (A and D not together) and designers from E, F. With A: the other engineer must be B, so H is in; the designer cannot be E, so F, and then the fifth member... A, B, F, H plus one of D/E/I — all forbidden (A–D, A–E, F–I) ✗. Without A: B and D, so H is in; the fifth is from E, F, I: BDEFH and BDEHI (BDFHI violates F–I). That gives 2 teams.
Total = 11 + 2 = 13 teams.
Teams containing I: ACGHI, BCGHI, CDEGI, CDGHI, BDEHI → 5. (F can never appear with I.)
Question 3
Single correct option
If H is not in the team, which of the following must be true?
- A. B is in the team
- B. E is in the team
- C. F is in the team
- D. A is in the team
Show answer and solution
Answer: (B) E is in the team
Listing the valid teams. Split by whether C and G (who come as a pair) are included.
*With C and G:* three more members, with at least one more engineer and at least one analyst. Checking all cases gives 11 teams: ABCGH, ACFGH, ACGHI, BCDGH, BCEGH, BCFGH, BCGHI, CDEGH, CDEGI, CDFGH, CDGHI.
*Without C and G:* engineers come from A, B, D (A and D not together) and designers from E, F. With A: the other engineer must be B, so H is in; the designer cannot be E, so F, and then the fifth member... A, B, F, H plus one of D/E/I — all forbidden (A–D, A–E, F–I) ✗. Without A: B and D, so H is in; the fifth is from E, F, I: BDEFH and BDEHI (BDFHI violates F–I). That gives 2 teams.
Total = 11 + 2 = 13 teams.
Without H, the analyst must be I, so F is out and B is out (B needs H). The only valid team is C, D, E, G, I. Hence E is in the team must be true; A, B and F are all excluded.
*With C and G:* three more members, with at least one more engineer and at least one analyst. Checking all cases gives 11 teams: ABCGH, ACFGH, ACGHI, BCDGH, BCEGH, BCFGH, BCGHI, CDEGH, CDEGI, CDFGH, CDGHI.
*Without C and G:* engineers come from A, B, D (A and D not together) and designers from E, F. With A: the other engineer must be B, so H is in; the designer cannot be E, so F, and then the fifth member... A, B, F, H plus one of D/E/I — all forbidden (A–D, A–E, F–I) ✗. Without A: B and D, so H is in; the fifth is from E, F, I: BDEFH and BDEHI (BDFHI violates F–I). That gives 2 teams.
Total = 11 + 2 = 13 teams.
Without H, the analyst must be I, so F is out and B is out (B needs H). The only valid team is C, D, E, G, I. Hence E is in the team must be true; A, B and F are all excluded.
Question 4
Single correct option
Which employee is a member of the largest number of possible teams?
- A. C
- B. G
- C. B
- D. H
Show answer and solution
Answer: (D) H
Listing the valid teams. Split by whether C and G (who come as a pair) are included.
*With C and G:* three more members, with at least one more engineer and at least one analyst. Checking all cases gives 11 teams: ABCGH, ACFGH, ACGHI, BCDGH, BCEGH, BCFGH, BCGHI, CDEGH, CDEGI, CDFGH, CDGHI.
*Without C and G:* engineers come from A, B, D (A and D not together) and designers from E, F. With A: the other engineer must be B, so H is in; the designer cannot be E, so F, and then the fifth member... A, B, F, H plus one of D/E/I — all forbidden (A–D, A–E, F–I) ✗. Without A: B and D, so H is in; the fifth is from E, F, I: BDEFH and BDEHI (BDFHI violates F–I). That gives 2 teams.
Total = 11 + 2 = 13 teams.
Counting appearances across the 13 teams: H is in 12 (every team except CDEGI), C and G are in 11 each, B and D in 7 each. So H appears most often.
*With C and G:* three more members, with at least one more engineer and at least one analyst. Checking all cases gives 11 teams: ABCGH, ACFGH, ACGHI, BCDGH, BCEGH, BCFGH, BCGHI, CDEGH, CDEGI, CDFGH, CDGHI.
*Without C and G:* engineers come from A, B, D (A and D not together) and designers from E, F. With A: the other engineer must be B, so H is in; the designer cannot be E, so F, and then the fifth member... A, B, F, H plus one of D/E/I — all forbidden (A–D, A–E, F–I) ✗. Without A: B and D, so H is in; the fifth is from E, F, I: BDEFH and BDEHI (BDFHI violates F–I). That gives 2 teams.
Total = 11 + 2 = 13 teams.
Counting appearances across the 13 teams: H is in 12 (every team except CDEGI), C and G are in 11 each, B and D in 7 each. So H appears most often.
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