I will assume that the number may not begin with 0 (zero).
The first digit cannot be zero: 9 choices.Find the number of 6-digit numbers that can be formed using the digits 0 to 9 if:
a) each digit must be different
The second digit can be any of the other 9 other digits.
The third digit can be any of the other 8 digits.
The fourth digit can be any of the other 7 digits.
The fifth digit can be any of the other 6 digits.
The sixth digit can be any of the other 5 digits.
The first digit has 9 choices.b) digits may be repeated
Each of the other five digits has 10 choices.
In part (b), we saw that there are possible numbers.c) at least one 5 is included
Consider the numbers that have no 5's.
The first digit has 8 choices (not 0 and not 5).
Each of the other five digits has 9 choices.
Hence, there are: . numbers with no 5's.
The number isd) the number is even, greater than or equal to 600,000, with repetition allowed
The first digit must be: 6, 7, 8, or 9 . . . 4 choices.
The number is even.
The last digit must be: 0, 2, 4, 6, 8 . . . 5 choices.
Each of the other four digits has 10 choices.