Learn / Library / Combinatorics

Couples on a panel

CombinatoricsHard

A panel of $5$ people is chosen from $6$ married couples ( $12$ distinct people). How many panels contain exactly one married couple, with no other couple fully included?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Pick the couple first, then pick the rest from different couples.

Solution

Choose the couple that is fully included: $6$ ways.
The remaining $3$ members must come from $3$ different couples among the other $5$ : $(3 5) = 10$ ways.
From each of those $3$ couples choose one person: $2^{3} = 8$ ways.
Multiply: $6 \times 10 \times 8$ .
Therefore the answer is $480$ .

Answer: $480$

Next question Back to the library