+275 How many three-digit multiples of 5 have three different digits?
How many three-digit multiples of 5 have three different digits and an odd tens digit?
+2668 I'll do number 1, and let you solve for number 2...
For number 1, there are 2 choices for the units digit (0 or 5), 9 choices for the tens digit (anything but the units digit), and 8 choices for the hundreds digit (anything but the tens or unit digit).
This makes for \(2 \times 9 \times 8 = 144\)choices, but we aren't done yet...
Our answer includes numbers like \(085\), which isn't a 3-digit number.
There is 1 choice for the hundreds digit, 8 choices for the tens digit, and 1 choice for the one's digit, which makes for \(1 \times 8 \times 1 = 8\) numbers.
So, there are \(144 - 8 = \color{brown}\boxed{136}\) numbers that work
+2668 There are 2 cases for number 2: the one digit is a 0, and the one's digit is a 5
For the former case: 1 choice for the one's digit, 5 choices for the 10s digit, and 8 choices for the hundreds digit, for \(8 \times 5 \times 1 = 40\)
For the latter case: 1 choice for the one's digit, 4 choices for the 10s digit (keep in mind we can't re-use 5), and 8 choices for the hundreds digit, for \(1 \times 4 \times 8 = 32\) numbers.
But, we overcount for 015, 035, 075, and 095, making the total \(40 + 32 - 6 = \color{brown}\boxed{68}\)
Note that it is the same as the previous answer divided by 2. Can you figure out why?
