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>(3,4)</math>. | ||
| + | |||
| + | All four pairs are asymmetrical; therefore by symmetry there are eight pairs altogether, so <math>\boxed{\rm{(B)}}</math>. | ||
| + | |||
| + | == Solution 2 == | ||
| + | Denote <math>T_n</math> as the number of such pairs for <math>n</math> people. Then for <math>T_{n-1}</math>, 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 <math>T_{n-1}</math>. Now if we note that <math>T_2=1</math>, we have that <math>T_5=8</math>, so that the answer is <math>\boxed{\rm{(B)8}}</math>. | ||
| + | |||
| + | == See also == | ||
| + | {{AHSME box|year=1989|num-b=23|num-a=25}} | ||
| + | |||
| + | [[Category: Intermediate Combinatorics Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 11:27, 19 June 2021
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 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
