Question

How many numbers greater than 1000, but not greater than 4000 can be formed with the digits 0, 1, 2, 3, 4, repetition of digits being allowed?

Answer 1

0, 1, 2, 3, 4. Five digits.
Numbers greater than 1000 and less than or equal to 4000 will be of four digits and will have either 1 (except 1000) or 2 or 3 in the first place or 4 in the first place with 0 in each of the remaining places.
Numbers having 1 in first place. After fixing 1st place, the second place can be filled by any of the 5 digits (not 4 digits because repetition is allowed i.e.1 can appear again). Similarly 3rd place can be filled up in 5 ways and 4th place can be filled in 5 ways. Thus there will be $5\times 5\times 5=125$ ways in which 1 will be in the first place. But this includes 1000 also which does not satisfy the given condition of being greater than 1000. Hence there will be 124 numbers having 1 in the first place. Similarly 125 each when 2 or 3 are in the first place. Only one number with 4 in the first place is formed, namely 4000 because of the condition of being less than or equal to 4000. Therefore total number of such numbers is
124 + 125 + 125 + 1 = 375.