Let a₁a₂a₃a₄a₅ be a 5 -element permutation formed using distinct elements from the set 1, 2, 3, , n where n 5 . It is given that there are exactly 666 such permutations in which either a₁, a₂, a₃ are consecutive integers in increasing order or a₃, a₄, a₅ are consecutive integers in increasing order. The value of n is

Step-by-step solution

Let A be the set of permutations where a₁, a₂, a₃ are consecutive integers in increasing order. The number of such 3 -element consecutive blocks from n integers is (n - 2) . The remaining 2 positions can be filled by the remaining (n - 3) elements in ^ (n-3) P₂ = (n - 3)(n - 4) ways. Thus, n(A) = (n - 2)(n - 3)(n - 4) . Let B be the set of permutations where a₃, a₄, a₅ are consecutive integers in increasing order. Similarly, n(B) = (n - 2)(n - 3)(n - 4) . The intersection A B represents permutations where all 5 ele