How many threedigit numbers can be formed under each conditi

How many three-digit numbers can be formed under each condition?

(a) The leading digit cannot be zero.
___numbers

(b) The leading digit cannot be zero and no repetition of digits is allowed.
numbers

(c) The leading digit cannot be zero and the number must be a multiple of 5.
numbers

(d) The number is at least 400.
numbers

Solution

(a) the leading digit cannot be zero
9 choices for first digit (1, 2, ..., 9)
10 choices for the second digit
10 choices for the third digit
9*10*10 = 900

(b) The leading digit cannot be zwro and no repetion of digits is allowed.
9 choices for first digit (1, 2, ..., 9)
9 choices for the second digit, anything different from the first
8 choices for the third digit, anything different from first and second
9*9*8 = 648

(c) The leading digit cannot be zero and the number is a multiple of 5
9 choices for first digit (1, 2, ..., 9)
10 choices for the second digit
2 choices for the third digit (0 or 5)
9*10*2 = 180

(d) the number is at least 400
6 choices for first digit (4, 5, ..., 9)
10 choices for the second digit
10 choices for the third digit
6*10*10 = 600