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If \(x%\) % of four-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is \(x\)? Express your answer as a decimal to the nearest tenth.

 Apr 26, 2018

Well if one repeats and the others are both different and this is a code so the first digit can be zero THEN


Pick 3 numbers out of 10 = 10C3 = 120

these can be arranged in 3! =6 ways


120*6= 720 3 digit numbers.

There are 3 possibilities for the 4 digit 

and the 4th digit can go in 4 different places.

but there are 2 the same so must divide by 2


so we have


120*6*3*4/2 = 4320 ways I think


there are 10^4 = 10000 four digit numbers if the first one can be zeros


so the prob of chosing one with just one digit that repeats is  4320/10000 = 0.432 = 43.2%



If ithe number cannot start with 0 then it would be different.  

I can work this out but it is quite a bit more complicated.

 Apr 26, 2018
edited by Melody  Apr 26, 2018

I do not completely understand this... How would it be less?

TheMathCoder  Apr 26, 2018

Please, if anyone can help and give a good explanation to back up their answer, I would greatly appreciate the help. Thank you!

 Apr 27, 2018

Did you understand any of it?

It would have been good if you had told me more specifically what you did not understand.


It is a 4 digit number but 1 digit (exactly one digit) is repeated just once.

So 3 digits out of 10 have to be chosen.

There are 10C3 = 120 ways to do this.

say the digits chose are  5,6 and 9, they can be in any order and there are 3! possible orders =6 orders.

So so far that is   120*6=720 ways to get a 3 digit number where the digits are all different.


But we want a 4 digit number.

One digit has to be repeated there are 3 possibilities so that is   720*3 = 2160 posibilities.


But there are 4 postions that extra number can be put in. So that is  2160*4=8640 numbers

But I have double counted because    1434 is the same if I swap the 4s around so I have to halve this answer


8640/2 = 4320    

So the number of favourable outcomes is 4320.


Altogether there are  10*10*10*10 = 10000 4 digit numbers (if they are allowed to start with 0)


so the prob of getting just one digit repeated once is   4320/10000 = 0.4320 = 43.2%


Do you understand now?

You will need to write your query on this thread but also send me a private message with the address of this thread attached. Otherwise I will not see it. 


If you do have a query try to ask a specific question.  

Anyway I hope that you understand :)

 Apr 27, 2018

Hi Mathcoder,


I see that you have viewed my answer, I am glad about that.    smiley

But you have not given me any written feedback.                        sad


I did ask specific questions of you. It would be nice of you to respond. 

 Apr 28, 2018

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