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# Counting and Probability

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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

#1
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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%

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

TheMathCoder  Apr 26, 2018
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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
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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
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Hi Mathcoder,