The number of 4 -digit numbers, strictly between 4000 and 7000 , that can be formed using the digits 0, 2, 3, 4, 5, 7 without repetition, such that they are divisible by 4 , is

Step-by-step solution

For a 4 -digit number to be strictly between 4000 and 7000 , its thousands digit must be 4 or 5 . For a number to be divisible by 4 , the number formed by its last two digits must be divisible by 4 . Using the digits 0, 2, 3, 4, 5, 7 , the possible 2 -digit endings divisible by 4 are 04, 20, 24, 32, 40, 52, 72 . Case 1: The thousands digit is 4 . Since repetition is not allowed, the ending cannot contain 4 . The valid endings are 20, 32, 52, 72 ( 4 choices). For each ending, 3 digits are used (the thousands digit a