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

Approach: Six people split into three labelled skills (Data, Design, Marketing), exactly 2 each. Instead of a giant enumeration, lock down C first (it is the most restricted), then count where the A–D split and the B–E–F rule can live. Symmetry between Data and Design halves the work.

Step 1: Pin C.
C \(\ne\) Marketing, so C is in Data or Design. By the Data\(\leftrightarrow\)Design symmetry, count the case C in Data fully and double it at the end.

Step 2: Read condition 2 the right way.
"B matches exactly one of E, F" means the trio \(\{B,E,F\}\) is not all in one skill and B is not alone away from both. Concretely, exactly one of E, F shares B's skill and the other does not. This is the key filter, so handle B, E, F together.

Step 3: Place the trio B, E, F.
With C fixed in Data, the two seats of each skill are: Data has 1 seat left, Design has 2, Marketing has 2. We need to seat B, E, F (3 people) plus A, D (2 people) into these 5 remaining seats. The clean way: choose B's skill, then force exactly one of \(\{E,F\}\) with B.

Step 4: Count by B's skill, applying A \(\ne\) D throughout.
Running the case split (B in Data / Design / Marketing, each time forcing exactly one of E,F with B and the rest filling the remaining two skills so that A and D end up in different skills) leaves exactly 8 valid full assignments when C is in Data. The capacity "2 per skill" and "A,D apart" together eliminate the rest.

Step 5: Use symmetry.
C in Design mirrors C in Data exactly (swap the Data and Design labels), giving another 8.

\[ 8 + 8 = 16 \]

Final Answer: \(\boxed{16}\)

Intuition: fixing the most-constrained element (C) and exploiting the Data–Design symmetry turns a 90-way raw count into two quick mirror-image counts.