A padlock has the digits 0 through 9 arranged in a circle on its face. A combination for 0 1 2 3 4 6 5 7 8 9 this padlock is four digits long. Because of the internal mechanics of the lock, no pair of consecutive numbers in the combination can be the same or one place apart on the face. For example 0-2-7-1 is a valid combination, but neither 0-4-4-7 (repeated digit 4) nor 3-0-9-5 (adjacent digits 0-9) are permitted. How many combinations are possible?

Answer:The total numbers of possible combinations are 3430.Step-by-step explanation:Consider the provided information.A combination for 0 1 2 3 4 6 5 7 8 9 this padlock is four digits long. Because of the internal mechanics of the lock, no pair of consecutive numbers in the combination can be the same or one place apart on the face.Here, for the first digit we have 10 choices.For the second digit we have 7 choices, as the digit can't be the same nor adjacent to the first digit.For the third digit we have 7 choices, as the digit can't be the same nor adjacent to the second digit.For the fourth digit we have 7 choices, as the digit can't be the same nor adjacent to the third digit.So the number of choices are:[tex]10\times 7\times 7\times 7=10\times 7^3\\10\times 343=3430[/tex]Hence, the total numbers of possible combinations are 3430.