Question

Six employees — A, B, C, D, E, F — each specialize in exactly one of three skills: Data, Design, or Marketing (two per skill).
1. A and D do not share a skill.
2. B’s skill is the same as either E or F (but not both).
3. C is not in Marketing.
How many valid assignments of skills are possible?

Hint:

Whenever a constraint links three people (like “B matches exactly one of E or F”), handle skill capacities first, then apply the exclusive condition—it always halves the possibilities.

Answer 1

Correct Answer: 16

Step 1: Assign A and D to two different skills: \( 3\times2=6 \) ordered ways. The one skill left over, used by neither A nor D, is the "open" skill.
Step 2: B cannot share a skill with A or D (the second condition only allows B's partner to be E or F), so B must go into the open skill, taking one of its two slots. Its partner there must be E or F: 2 choices.
Step 3: The two people left over, C and whichever of E/F was not chosen, must now fill the single remaining slot in A's skill and the single remaining slot in D's skill. Since C cannot enter Marketing, this forces a specific placement whenever Marketing happens to be A's skill or D's skill, but allows either placement when Marketing is instead the open skill.
Step 4: When the open skill is Marketing (2 of the 6 A/D-assignments), both arrangements of the last two people are valid, giving \( 2\times2\times2=8 \). When Marketing is A's skill or D's skill (4 of the 6 A/D-assignments), only one arrangement avoids putting C in Marketing, giving \( 4\times2\times1=8 \).
\[ \boxed{8+8=16} \]