All 6 -digit numbers formed using the digits 1, 2, 3, 4, 5, 6 without repetition are arranged in increasing dictionary order. The remainder when the 450^ th number in this sequence is divided by 13 is

Step-by-step solution

The total number of 6 -digit numbers is 6! = 720 . The number of numbers starting with 1, 2, 3, 4, 5, 6 is 5! = 120 each. Numbers starting with 1 : 120 Numbers starting with 2 : 120 Numbers starting with 3 : 120 Total numbers starting with 1, 2 , or 3 = 360 . The 450^ th number must start with 4 . We need the (450 - 360) = 90^ th number starting with 4 . For numbers starting with 4 , the remaining digits are 1, 2, 3, 5, 6 . Numbers starting with 41 : 4! = 24 Numbers starting with 42 : 24 Numbers starting with 43 :