# Number of $4$ digit numbers with no repeated digit.

Number of $4$ digit numbers with no repeated digit is

1. $4536$
2. $3024$
3. $5040$
4. $4823$

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Hi: with the level of information of your own work given here, most people are going to think that your "please help" means "please do it for me". If you include your thoughts up to this point, people will be more likely (and able) to give you helpful information. Add the homework tag too, if that is the case. –  rschwieb Jun 11 '12 at 11:35
Hello, this is not a home work.well, I will definitely improve my thinking on combinotorics. –  La Belle Noiseuse Jun 11 '12 at 11:47

Ok so lets write down any old $4$ digit number

$abcd$

How many choices do we have for the digit $a$? We have $9$ choices (since the first digit cannot be $0$). Now for each possible choice of $a$ we have $9$ choices for $b$ (since we want $b$ to be a different digit to $a$ and we now allow $0$).

So for choice of the $ab$ part we have $9*9 = 81$ possibilities.

Now for each of these we have $8$ choices for $c$ (to avoid $c$ being the same as either $a$ or $b$). And for each of these we have $7$ choices for $d$ (to avoid $d$ being the same as either $a,b$ or $c$).

So in total there are $9*9*8*7 = 4536$ possible numbers.

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ah! so beautiful answer thank you fretty. –  La Belle Noiseuse Jun 11 '12 at 11:45
I can't help but notice that there are no repeated digits in the number $4536$. Somehow the answer to the question was among the numbers being counted, how mysterious! –  Marc van Leeuwen Jun 11 '12 at 13:11

A bit more hint: Ask yourself how many ways can you choose the digit in the first position, then how many ways can you choose the second-place digit ...

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Hint: this is like drawing four digits out of a bag of ten digits 0-9. How many different ways can this happen?

Edit: To correct the model I had in mind, the bag would first have to contain only 1-9. After you've picked the leftmost digit, you would throw in a 0 ball and continue to pick the last three digits.

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Not exactly, since the choice of the first digit is restricted. –  fretty Jun 11 '12 at 11:37
@fretty True! A scurvily written question. –  rschwieb Jun 11 '12 at 11:42