Hello, william!

I will assume that the number maynotbegin 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.

Answer: .

The first digit has 9 choices.b) digits may be repeated

Each of the other five digits has 10 choices.

Answer: .

In part (b), we saw that there are possible numbers.c) at least one 5 is included

Consider the numbers that haveno5'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.

Answer: .

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.

Answer: .