The number of four digit numbers that can be formed from the digits 0, 1, 2, 3, 4, 5 with at least one digit repeated is

The thousand’s place cannot be filled by O. So the number of ways to fill the thousands’s place = 5.The remaining three places can be filled by six digits in 63 ways, as digits can be repeated.∴ The number of four digit numbers= 5 × 63 = 1080.                         (1)Now, the number of numbers of four digits that do not contain any repeated digit =5P1×5P3 {∵ thousand’s place is to be filled by one of 1, 2, 3, 4, 5 and the remaining three places are to be filled by three of the remaining five digits including 0}. But (1) contains numbers which contain no repeated digits as well as those which contain at least one repeated digit.∴ The number of four digit numbers which contain at least one repeated digit =1080−5P1×5P3= 1080 – 5 × 5 × 4 × 3 = 780.