1989 AHSME Problems/Problem 24
Contents
Problem
Five people are sitting at a round table. Let be the number of people sitting next to at least 1 female and be the number of people sitting next to at least one male. The number of possible ordered pairs is
Solution
Suppose there are more men than women; then there are between zero and two women.
If there are no women, the pair is . If there is one woman, the pair is .
If there are two women, there are two arrangements: one in which they are together, and one in which they are apart, giving the pairs and .
All four pairs are asymmetrical; therefore by symmetry there are eight pairs altogether, so .
Solution 2
Denote as the number of such pairs for people. Then for , when we add an extra spot, we can either have a male or female giving two options. Note that these two options however double the value of . Now if we note that , we have that , so that the answer is .
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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