Five-digit numbers are formed using the digits 1, 2, 3, , 9 without repetition. The total number of such five-digit numbers, in which either the first three digits are consecutive integers in increasing order or the last three digits are consecutive integers in increasing order, is

Step-by-step solution

Let the five-digit number be d₁ d₂ d₃ d₄ d₅ . Let A be the set of numbers where the first three digits d₁, d₂, d₃ are consecutive integers in increasing order. The possible consecutive blocks of three digits from 1, 2, , 9 are (1,2,3), (2,3,4), , (7,8,9) . There are 7 such blocks. For each block, the remaining 2 positions can be filled by the remaining 6 digits in ⁶P₂ = 30 ways. Thus, n(A) = 7 30 = 210 . Let B be the set of numbers where the last three digits d₃, d₄, d₅ are consecutive integers in increasing order.