Big combinations doubt

Question

Hi guys!

So I've been working on combinations and one place I always get stuck in, is knowing when to further arrange the combinations. 
To explain. Here's a sum I just solved

1. How many positive integers of three different digits, each less than 400 can be formed from the digits 1, 2, 3, 4, 5 and 6?

So, the way to solve this would be - 3*(5*4) = 60 ways 
But my question here is, if the order of the tens digit and unit's digit don't matter, shouldn't we further multiply the answer by 2! to arrange the tens and unit's digit?

Am I missing something here?

logro · Answer

Hey Sinking boat,

In short you don't need to multiply it by 2! since the 60 ways include the re-arrangement possibility

For the hundreds digit, the possible numbers are from 1,2,3, select 1 from these which is 3C1

For the tens digit, since it has to be a unique number you cannot select the number which is in the hundreds digit so you can do it from 5 remaning (6 numbers - the hundred's number)

Similarly you can choose from 4 numbers for the units digit

Now take the case of 235 and 253

our units digit is fixed at 2

for the tens digit when we say there are 5 numbers you can pick from 1,3,4,5 and 6 (Both 3 and 5 are covered)

Similarly for the units digit

Kudos if it helps. Happy to explain further if you have any questions

sinkingbot · Answer

Thanks for explaining! 
So how do i know when i need to further arrange the answer?

IanStewart · Answer

If you're creating a 3-digit number, the order of the digits definitely matters. The number '251' is different from the number '215' precisely because of the order of the digits.

From the way you've described your thought process, I'm guessing you might have studied counting from a book that teaches you to select your items first, then multiply by the possible number of arrangements of those items in the case that order matters. That's an especially confusing way to think about counting when order matters, so I wouldn't recommend learning the subject that way (if you have been studying from a source that explains counting that way, you might want to try a book that explains things more simply).

When order matters (as will always be the case when you're making numbers, words, passwords, license plates, seating arrangements, or filling labeled positions, among other situations), all you need to do is count how many choices you have for each thing, and multiply. In your digits question, you have 3 choices for the first digit, then 5 for the tens digit, and 4 for the ones digit. So the answer is (3)(5)(4) = 60.

Big combinations doubt

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