Consider a permutation (a_1, a_2, a_3, a_4, a_5) of \ 1,2,3,4,5\ . We say the 5-tuple (a_1, a_2, a_3, a_4, a_5) is flawless if for all 1 ≤ i

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Consider a permutation (a_1, a_2, a_3, a_4, a_5) of \ 1,2,3,4,5\ . We say the 5-tuple (a_1, a_2, a_3, a_4, a_5) is flawless if for all 1 ≤ i < j < k ≤ 5 , the sequence (a_j, a_j, a_k) is not an arithmetic progression

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Correct answer: 19

There are only four possible three terms AP exist, which are , , and Consider the sets A= are present in the sequence B= are present in the sequence C= are present in the sequence D= are present in the sequence aligned & n(A)=n(B)=n(C)=n(D)= ^5 C_3 · 2 · 2 =40 & n(A B)=n(B C)= ^5 C_4 · 2=10 & n(A C)=1+6 × 2=13 & n(A D)= ^5 C_4 · 2=n(C D)=10 & n(B D)=4 ·|2| 2=16 & n(A B C)=2 & n(B C D)=4 & n(A C D)=2 & n(A B D)=4 and n(A B C D)=2 & Now n(A B C D)=4(40)-69+12-2=101 & So number of flawless 5-tuples =5-101=19 aligned

Consider a permutation (a_1, a_2, a_3, a_4, a_5) of \ 1,2,3,4,5\ ....

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