The number of 4-digit numbers in the closed interval [2024, 4000] by using the digits 0, 2, 3,4,6,7 is n, then number of digits in n are 

Case  i)     2 0 2 4,2026,2027(three numbers)

Case ii) four digit numbers that begin with 20-- third place (3,4,6,7) and last  place can be(0,2,3,4,6,7)
 =24

 3
 Case iii)  number of 4 digit numbers begin with 2 and second place with 2,3,4,6,7 and (last two places can be filled with 0,2,3,4,6,7)
 is 5X36=180

 Case iv)   number of 4 digit numbers begin with   3 - - - =6³=216
 Case v)      4 0 0 0 =1 
 Total number of ways =3+24+180+216+1=424