1989 AMC 12 Problem 24

Question

Five people are sitting at a round table. Let

f\ge0

be the number of people sitting next to at least one female and

m\ge0

be the number of people sitting next to at least one male. The number of possible ordered pairs

(f,m)

is

7

8

9

10

11

Classify arrangements by the number of females

With two females, separate the adjacent and nonadjacent cases; obtain three-female cases by symmetry

For

0,1,2

females, the possible pairs are respectively

\begin{aligned} &(0,5),\ (2,5),\\ &(4,5),\ (3,4), \end{aligned}

where the two-female cases distinguish adjacent from nonadjacent females. Swapping the sexes gives

\begin{aligned} &(5,0),\ (5,2),\\ &(5,4),\ (4,3). \end{aligned}

These are

8

distinct ordered pairs.

Thus the correct answer is B.

Problem 24 in other years