[jacks]

Let \( a_{1},a_{2},a_{3}......,a_{10} \) be a permutation of the set \( \left\{1,2,3......,10 \right\} \) such that the sequence \( a_{i} \)

decreases first and then increases like \( 8,6,4,1,2,3,5,7,9,10 \). If \( N \) is the number of such permutations,then the sum of digits of \( N \)

is

[Luis Fuentes]

Hello

 I suppose that permutations \( \{1,2,3,4,5,6,7,8,9,10\} \) and \( \{10,9,8,7,6,5,4,3,2,1\} \) are excluded.

 To construct a permutation under the prescribed conditions we only have to choose the numbers of the decreasing part.

 For example if we choose \( 2,9,4,5 \), we obtain the permutation \( \{9,5,4,2,1,3,6,7,8,10\} \).

 Thus, the number of such permutations is the number of non-trivial subsets of \( \{1,2,3,4,5,6,7,8,9,10\} \).

 Conclude...

Best regards