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# A popular website requires users to create a password consis

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25 Nov 2010, 18:49
2
15
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Difficulty:

55% (hard)

Question Stats:

60% (01:52) correct 40% (02:05) wrong based on 206 sessions

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A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

A. 9! + 10!
B. 2 x 10!
C. 9! x 10!
D. 19!
E. 20!

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I'm the Dumbest of All !!
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26 Nov 2010, 10:26
5
5
shrive555 wrote:
A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

9! + 10!
2 x 10!
9! x 10!
19!
20!

i got it right with slot method, but wasn't sure.

shrive555 wrote:
why 9! is multiplied with 10 ?

thanks

Note that we are told that no digit may be repeated.

There are total of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

As password must be at least 9 digits long then it can consist of either 9 or all 10 digits.

If password consists of 9 digits then # of passwords possible is $$C^9_{10}*9!=10*9!=10!$$: $$C^9_{10}$$ - # of ways to choose 9 digits which will be used in password and $$9!$$ arranging these digits (or in another way $$P^9_{10}=10!$$ - choosing 9 digits out of 10 when the order matters);

If password consists of all 10 digits then # of passwords possible is simply $$10!$$;

So total # of passwords possible is $$10!+10!=2*10!$$.

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##### General Discussion
Director
Joined: 03 Sep 2006
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25 Nov 2010, 19:48
2
1
If we choose the 10 different digits then they can be arranged (permutations) in 10! ways.
But the question asks at least 9 digits. So we have the possibility of choosing only 9 digits for the password ( but digit shouldn't repeat), so we can have a total of 10 different combinations and each combination can be arranged in 9! ways.
Therefore
10 x 9! + 10!

= 10! + 10!

= 2 x 10!

Therefore B
Manager
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26 Nov 2010, 10:16
why 9! is multiplied with 10 ?

thanks
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26 Nov 2010, 11:07
Got it.
012345678 is one password 123456789 is another, by combination method we can find how many arrangements of 9 digits are possible, which is 10
Thanks B
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26 Nov 2010, 12:42
It's good to realize the permutation formula: n! / (n-k)!

We then got:

10! / (10-9)! = 10!/1! = 10!
and
10! / (10-10)! = 10!/0! = 10! = 10!/1! = 10! (

The answer is than: 2 * 10!
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Re: A popular website requires users to create a password consis  [#permalink]

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15 May 2017, 21:39
Would have been brutal if 10! would have been one of the answers. It's easy to ignore the word atleast..
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Re: A popular website requires users to create a password consis  [#permalink]

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12 Jun 2017, 01:40
shrive555 wrote:
A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

A. 9! + 10!
B. 2 x 10!
C. 9! x 10!
D. 19!
E. 20!

The question must mention that the password can start with a '0'. Bunuel, if my reasoning is correct, would you please edit the question?
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Re: A popular website requires users to create a password consis  [#permalink]

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12 Jun 2017, 02:43
ShashankDave wrote:
shrive555 wrote:
A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

A. 9! + 10!
B. 2 x 10!
C. 9! x 10!
D. 19!
E. 20!

The question must mention that the password can start with a '0'. Bunuel, if my reasoning is correct, would you please edit the question?

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12 Jun 2017, 09:56
9 digit passwords = $$10_C_9$$ * 9! = 10!
10 digit passwords = $$10_C_{10}$$ * 10!= 10!
sum = 2*10!
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Re: A popular website requires users to create a password consis  [#permalink]

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13 Jun 2017, 03:41
praneet87 wrote:
Would have been brutal if 10! would have been one of the answers. It's easy to ignore the word atleast..

Couldn't agree more.

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Re: A popular website requires users to create a password consis  [#permalink]

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14 Jun 2017, 16:26
shrive555 wrote:
A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

A. 9! + 10!
B. 2 x 10!
C. 9! x 10!
D. 19!
E. 20!

The number of ways to create a 10-digit password from 10 digits is 10P10 = 10!.

Thus, the number of ways to create the codes is 10! + 10! = 2(10!).

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Re: A popular website requires users to create a password consis  [#permalink]

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22 Mar 2019, 02:45
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Re: A popular website requires users to create a password consis   [#permalink] 22 Mar 2019, 02:45
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