Problem Solving! Watch

melissadh
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I have bought a combination lock for my bicycle. The combination is a three digit number in the range 000-999 and it is currently set to the manufacturer's default combination of 999. Before I use the lock I need to set the combination. I want to use a combination where all three digits are different. How many combinations are there in the range 000-999 where all three digits are different?

I got close to the answer 720, but I got 729 instead. Why are 9 combinations lost? How did you get to the ANS?

Thank you so much!
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annabel_gilmour
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Not sure how you got 729 but here is how to get 720!

There are 3 digits, and we know all three must be different. A digit can be any number from 0-9. This gives us 10 different numbers each digit could be in theory, if there weren't any limitations over each digit. The first digit can be any of the numbers 0-9.

So for the first digit there are 10 possible options.

For the second digit we know that it can't be the same as the first one, but can still be any of the other numbers. So we take 1 away from 10 to leave us with 9 possible digits, as one of the original 10 has already been used, and we know that the same digit can not be used to twice.

So for the second digit there are 9 possible options.

For the last digit, we know that this can't be the same as either of the digits used before. So since we've used two, this takes two possible options away from the original 10, leaving us with 8 possible options the digit could be.

So for the third digit there are 8 possible options.

Using the product rule for counting (if there are m ways to do one thing, n ways to do another, and l ways to do another, there are mnl ways to do all three) we multiply 10 by 9 by 8 to get 720, so there are 720 possible combinations!
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RDKGames
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(Original post by melissadh)
I have bought a combination lock for my bicycle. The combination is a three digit number in the range 000-999 and it is currently set to the manufacturer's default combination of 999. Before I use the lock I need to set the combination. I want to use a combination where all three digits are different. How many combinations are there in the range 000-999 where all three digits are different?

I got close to the answer 720, but I got 729 instead. Why are 9 combinations lost? How did you get to the ANS?

Thank you so much!
There are 10 digits to choose from in the first slot.
Then there are 9 to choose from in the second slot.
Then there are 8 to choose from in the third slot.
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melissadh
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(Original post by RDKGames)There are 10 digits to choose from in the first slot.
Then there are 9 to choose from in the second slot.
Then there are 8 to choose from in the third slot.

Thanks guys! I can't believe it was as simple as that. Just after I posted this, I got the answer using a different method, but this, too, makes sense.
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