Difference between revisions of "2025 SSMO Accuracy Round Problems/Problem 8"
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==Problem== | ==Problem== | ||
| − | We say that a permutation <math>(a_1, a_2, \dots ,a_{10})</math> of the integers <math>1</math> through <math>10</math> inclusive is | + | We say that a permutation <math>(a_1, a_2, \dots ,a_{10})</math> of the integers <math>1</math> through <math>10</math> inclusive is <i>peaked</i> if there do not exist three integers <math>1\le i < j < k \le 10</math> such that <math>a_i > a_j </math> and <math>a_j< a_k</math>. Let <math>\mathcal{S}</math> be the set of all peaked permutations. If <math>a_p = 9</math> and <math>a_q = 4</math>, the expected value of <math>|p-q|</math> over all permutations in <math>\mathcal{S}</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is the value of <math>m+n</math>? |
==Solution== | ==Solution== | ||
Latest revision as of 11:27, 9 September 2025
Problem
We say that a permutation 
of the integers 
through 
inclusive is peaked if there do not exist three integers 
such that 
and 
. Let 
be the set of all peaked permutations. If 
and 
, the expected value of 
over all permutations in can be written as 
, where 
and 
are relatively prime positive integers. What is the value of 
?
