Final answer:
To find the arrangements of the numbers 1 through 6 such that any two adjacent numbers have an even product, position the even and odd numbers alternately. This results in 3! × 3!, or 36, possible arrangements.
Step-by-step explanation:
To find the number of ways of arranging the numbers 1, 2, 3, 4, 5, 6 so that the product of any two adjacent numbers is even, we must recognize that an even product occurs if at least one of the numbers in the pair is even. In the given set, the even numbers are 2, 4, and 6. For an arrangement to satisfy the condition, there can be no two odd numbers (1, 3, 5) adjacent to each other.
To achieve this, we can start by arranging the even numbers and then place the odd numbers in the remaining spaces. There are 3! (3 factorial) ways to arrange the even numbers and 3! ways to arrange the odd numbers in between. Therefore, we have 3! × 3! ways to arrange the even numbers and the odd numbers separately, giving us a total of 6 × 6 = 36 possible arrangements where the product of any two adjacent numbers is even.