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

Step 1: The tightest rule is "F not in P1," so fix P1's makeup first: its two seats must come from A, B, C, D, E only.
Step 2: B needs C or F alongside; since F cannot be in P1, any P1 slot containing B forces C into P1 as well, and every other choice for P1 pushes B's C/F partner into P2 or P3 instead.
Step 3: Layering the A–D separation and the E-not-with-C rule on top of each P1 choice, and keeping only the combinations that survive all four rules, the tally comes to
\[ \boxed{12} \]
Final Answer: 12.