Difference between revisions of "1989 AHSME Problems/Problem 24"
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+ | == Problem == | ||
+ | |||
Five people are sitting at a round table. Let <math>f\geq 0</math> be the number of people sitting next to at least 1 female and <math>m\geq0</math> be the number of people sitting next to at least one male. The number of possible ordered pairs <math>(f,m)</math> is | Five people are sitting at a round table. Let <math>f\geq 0</math> be the number of people sitting next to at least 1 female and <math>m\geq0</math> be the number of people sitting next to at least one male. The number of possible ordered pairs <math>(f,m)</math> is | ||
<math> \mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 11 } </math> | <math> \mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 11 } </math> | ||
+ | |||
+ | == Solution == | ||
+ | Suppose there are more men than women; then there are between zero and two women. | ||
+ | |||
+ | If there are no women, the pair is <math>(0,5)</math>. If there is one woman, the pair is <math>(2,5)</math>. | ||
+ | |||
+ | 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 <math>(4,5)</math> and <math>(3,5)</math>. | ||
+ | |||
+ | All four pairs are asymmetrical; therefore by symmetry there are eight pairs altogether, so <math>\rm{(B)}</math>. | ||
+ | |||
+ | == See also == | ||
+ | {{AHSME box|year=1989|num-b=23|num-a=25}} | ||
+ | |||
+ | [[Category: Intermediate Combinatorics Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 02:20, 4 February 2016
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 .
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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