Difference between revisions of "2008 AMC 12A Problems/Problem 21"
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<math>\mathrm{(A)}\ 36\qquad\mathrm{(B)}\ 40\qquad\textbf{(C)}\ 44\qquad\mathrm{(D)}\ 48\qquad\mathrm{(E)}\ 52</math> | <math>\mathrm{(A)}\ 36\qquad\mathrm{(B)}\ 40\qquad\textbf{(C)}\ 44\qquad\mathrm{(D)}\ 48\qquad\mathrm{(E)}\ 52</math> | ||
− | ==Solution== | + | ==Solution 1 (Symmetry) == |
There are <math>5!=120</math> total permutations. | There are <math>5!=120</math> total permutations. | ||
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There are <math>3</math> combinations of numbers, <math>2</math> possibilities of which side of the equation is <math>a_1+a_2</math> and which side is <math>a_4+a_5</math>, and <math>2^2=4</math> possibilities for rearranging <math>a_1,a_2</math> and <math>a_4,a_5</math>. Thus, there are <math>3\cdot2\cdot4=24</math> permutations such that <math>a_1 + a_2 = a_4 + a_5</math>. | There are <math>3</math> combinations of numbers, <math>2</math> possibilities of which side of the equation is <math>a_1+a_2</math> and which side is <math>a_4+a_5</math>, and <math>2^2=4</math> possibilities for rearranging <math>a_1,a_2</math> and <math>a_4,a_5</math>. Thus, there are <math>3\cdot2\cdot4=24</math> permutations such that <math>a_1 + a_2 = a_4 + a_5</math>. | ||
− | Thus, the number of <u>heavy-tailed</u> permutations is <math>\frac{120-24}{2}=48 \Rightarrow D</math>. | + | Thus, the number of <u>heavy-tailed</u> permutations is <math>\frac{120-24}{2}=48 \Rightarrow D</math>. |
+ | |||
+ | ==Solution 2 (Casework)== | ||
+ | We use case work on the value of <math>a_3</math>. | ||
+ | |||
+ | Case 1: <math>a_3 = 1</math>. Since <math>a_1 + a_2 < a_4 + a_5</math>, <math>(a_1, a_2)</math> can only be a permutation of <math>(2, 3)</math> or <math>(2, 4)</math>. The values of <math>a_1</math> and <math>a_2</math>, as well as the values of <math>a_4</math> and <math>a_5</math>, are interchangeable, so this case produces a total of <math>2(2 \cdot 2) = 8</math> solutions. | ||
+ | |||
+ | Case 2: <math>a_3 = 2</math>. Similarly, we have <math>(a_1, a_2)</math> is a permutation of <math>(1, 3)</math>, <math>(1, 4)</math>, or <math>(1, 5)</math>, which gives a total of <math>3(2 \cdot 2) = 12</math> solutions. | ||
+ | |||
+ | Case 3: <math>a_3 = 3</math>. <math>(a_1, a_2)</math> is a permutation of <math>(1, 2)</math> or <math>(1, 4)</math>, which gives a total of <math>2(2 \cdot 2) = 8</math> solutions. | ||
+ | |||
+ | Case 4: <math>a_3 = 4</math>. <math>(a_1, a_2)</math> is a permutation of <math>(1, 2)</math>, <math>(1, 3)</math>, or <math>(2, 3)</math>, which gives a total of <math>3(2 \cdot 2) = 12</math> solutions. | ||
+ | |||
+ | Case 5: <math>a_3 = 5</math>. <math>(a_1, a_2)</math> is a permutation of <math>(1, 2)</math> or <math>(1, 3)</math>, which gives a total of <math>2(2 \cdot 2) = 8</math> solutions. | ||
+ | |||
+ | Therefore, our answer is <math>8 + 12 + 8 + 12 + 8 = 48 \Rightarrow \boxed{D}</math>. | ||
+ | |||
+ | -MP8148 | ||
== See also == | == See also == | ||
{{AMC12 box|year=2008|num-b=20|num-a=22|ab=A}} | {{AMC12 box|year=2008|num-b=20|num-a=22|ab=A}} | ||
+ | |||
+ | [[Category:Introductory Combinatorics Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 00:21, 31 July 2023
Problem
A permutation of is heavy-tailed if . What is the number of heavy-tailed permutations?
Solution 1 (Symmetry)
There are total permutations.
For every permutation such that , there is exactly one permutation such that . Thus it suffices to count the permutations such that .
, , and are the only combinations of numbers that can satisfy .
There are combinations of numbers, possibilities of which side of the equation is and which side is , and possibilities for rearranging and . Thus, there are permutations such that .
Thus, the number of heavy-tailed permutations is .
Solution 2 (Casework)
We use case work on the value of .
Case 1: . Since , can only be a permutation of or . The values of and , as well as the values of and , are interchangeable, so this case produces a total of solutions.
Case 2: . Similarly, we have is a permutation of , , or , which gives a total of solutions.
Case 3: . is a permutation of or , which gives a total of solutions.
Case 4: . is a permutation of , , or , which gives a total of solutions.
Case 5: . is a permutation of or , which gives a total of solutions.
Therefore, our answer is .
-MP8148
See also
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 | |
All AMC 12 Problems and Solutions |
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