Five distinct natural numbers are chosen from the set 1, 2, 3, , 20 and arranged in strictly increasing order. If the middle (third) number is a multiple of 6 , then the total number of ways to form such an arrangement is :

Step-by-step solution

Let the five chosen numbers be a₁, a₂, a₃, a₄, a₅ such that a₁ The middle number a₃ is a multiple of 6 . The possible multiples of 6 in the set 1, 2, , 20 are 6, 12 , and 18 . Case 1: a₃ = 6 The two numbers a₁, a₂ must be selected from 1, 2, 3, 4, 5 . Number of ways = ⁵C₂ = 10 . The two numbers a₄, a₅ must be selected from 7, 8, , 20 (14 numbers). Number of ways = ¹⁴C₂ = 91 . Number of ways for this case = 10 91 = 910 . Case 2: a₃ = 12 The two numbers a₁, a₂ must be selected from 1, 2, , 11 . Number of ways = ¹¹C₂