Question 118
Let $p = (a_1; a_2; \dots; a_{12})$ be a permutation of the numbers $1, 2, \dots, 12$. Denote by $S_p = |a_1 - a_2| + |a_2 - a_3| + \dots + |a_{11} - a_{12}|$. We will call $p$ "optimistic", if $a_i \gt \min(a_{i-1}; a_{i+1})$ for every $i = 2, \dots, 11$.
a) Determine the maximal possible value of $S_p$? For how many permutations $p$ the maximum is achieved?
b) Find the number of all optimistic $p$?
c) Determine the maximal possible value of $S_p$ for an optimistic $p$? For how many optimistic $p$ the maximum is achieved?
a) Determine the maximal possible value of $S_p$? For how many permutations $p$ the maximum is achieved?
b) Find the number of all optimistic $p$?
c) Determine the maximal possible value of $S_p$ for an optimistic $p$? For how many optimistic $p$ the maximum is achieved?