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0
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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?

Jul 26, 2022

#1
+1

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

Jul 26, 2022
#2
0

I got 72 for the second one but it is in correct
2*5*8 = 80
80-8
72

Jul 26, 2022
#3
+1

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?

BuilderBoi  Jul 26, 2022