Difference between revisions of "2008 AMC 12A Problems/Problem 21"
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A permutation <math>(a_1,a_2,a_3,a_4,a_5)</math> of <math>(1,2,3,4,5)</math> is <u>heavy-tailed</u> if <math>a_1 + a_2 < a_4 + a_5</math>. What is the number of heavy-tailed permutations? | A permutation <math>(a_1,a_2,a_3,a_4,a_5)</math> of <math>(1,2,3,4,5)</math> is <u>heavy-tailed</u> if <math>a_1 + a_2 < a_4 + a_5</math>. What is the number of heavy-tailed permutations? | ||
| − | <math>\ | + | <math>\mathrm{(A)}\ 36\qquad\mathrm{(B)}\ 40\qquad\textbf{(C)}\ 44\qquad\mathrm{(D)}\ 48\qquad\mathrm{(E)}\ 52</math> |
==Solution== | ==Solution== | ||
Revision as of 00:55, 26 April 2008
Problem
A permutation 
of 
is heavy-tailed if 
. What is the number of heavy-tailed permutations?
Solution
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 
.
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 | |
