The total number of 5 -digit numbers, formed by using the digits 1, 2, 3, 4, 5, 6, 7 without repetition, which are multiples of 36 , is

Step-by-step solution

For a number to be a multiple of 36 , it must be divisible by both 9 and 4 . The sum of all 7 given digits is 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 . When forming a 5 -digit number, 2 digits are excluded. The sum of the 5 chosen digits must be a multiple of 9 . The minimum possible sum of 5 digits is 1 + 2 + 3 + 4 + 5 = 15 , and the maximum is 3 + 4 + 5 + 6 + 7 = 25 . The only multiple of 9 in this range is 18 . Thus, the sum of the 2 excluded digits must be 28 - 18 = 10 . The valid pairs of digits from the given set that