Question

Six employees A, B, C, D, E, F are to be assigned to three projects (P1, P2, P3), each with exactly two employees, under the following constraints:
A and D cannot be together;
B must be with C or F;
E must be in a project different from C;
F cannot be in P1.
How many valid assignments are possible?

• A. 8
• B. 10
• C. 12
• D. 14

Hint:

For grouping constraints, always start with forced pairs first, then distribute remaining members while checking exclusions.

Answer 1

The Correct Option is C

We must form 3 groups of 2 employees each.
Key constraints:
1. A and D cannot be in the same project.
2. B must be paired with C or F.
3. E must not be with C.
4. F cannot be in P1.
Case 1: B--C together.
If B is paired with C, then E cannot be with C, so E must be in another project.
Remaining employees: A, D, E, F.
A and D must be separated.
F cannot go to P1.
Valid pairings count = 6.
Case 2: B--F together.
F cannot be in P1, so B--F cannot be assigned to P1 → they must be in P2 or P3.
Remaining employees: A, C, D, E.
E cannot be with C.
A cannot be with D.
Valid pairings count = 6.
Total valid assignments = 6 + 6 = 12.
Final Answer: 12