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# A “descending number” is a three-digit number, such that the units dig

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Math Expert
Joined: 02 Sep 2009
Posts: 46991
A “descending number” is a three-digit number, such that the units dig [#permalink]

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15 Aug 2017, 23:33
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Difficulty:

95% (hard)

Question Stats:

44% (03:09) correct 56% (02:50) wrong based on 77 sessions

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A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25

(B) 2/9

(C) 2/15

(D) 1/9

(E) 1/10

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Joined: 09 Mar 2016
Posts: 648
Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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16 Aug 2017, 00:59
[quote="Bunuel"]A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25

(B) 2/9

(C) 2/15

(D) 1/9

(E) 1/10

hello Bunuel how are you hope you are well!:)

you know I tried to solve your questions but arrived at totally different answer

the descending numbers are as follow:

210, 321, 432, 543, 654, 765, 876, 987, 320, 430, 540, 650, 760, 870, 980 total 15 numbers

the total number of three digit numbers is 999-100+1= 900

the probability that descending number is picked at random is 15/900 = reducing it I get 1/60

can you please say where I am wrong ? thank you !
Intern
Joined: 28 May 2017
Posts: 18
Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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16 Aug 2017, 01:10
dave13 wrote:
Bunuel wrote:
A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25

(B) 2/9

(C) 2/15

(D) 1/9

(E) 1/10

hello Bunuel how are you hope you are well!:)

you know I tried to solve your questions but arrived at totally different answer

the descending numbers are as follow:

210, 321, 432, 543, 654, 765, 876, 987, 320, 430, 540, 650, 760, 870, 980 total 15 numbers

the total number of three digit numbers is 999-100+1= 900

the probability that descending number is picked at random is 15/900 = reducing it I get 1/60

can you please say where I am wrong ? thank you !
you forget to consider the nos like 310 , 431,421,430,420,410,530,530,540 etc and many more that's why you came up with a wrong answer

Sent from my iPhone using GMAT Club Forum mobile app
Director
Joined: 09 Mar 2016
Posts: 648
A “descending number” is a three-digit number, such that the units dig [#permalink]

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16 Aug 2017, 01:24
Sahilpandey wrote:
dave13 wrote:
Bunuel wrote:
A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25

(B) 2/9

(C) 2/15

(D) 1/9

(E) 1/10

hello Bunuel how are you hope you are well!:)

you know I tried to solve your questions but arrived at totally different answer

the descending numbers are as follow:

210, 321, 432, 543, 654, 765, 876, 987, 320, 430, 540, 650, 760, 870, 980 total 15 numbers

the total number of three digit numbers is 999-100+1= 900

the probability that descending number is picked at random is 15/900 = reducing it I get 1/60

can you please say where I am wrong ? thank you !
you forget to consider the nos like 310 , 431,421,430,420,410,530,530,540 etc and many more that's why you came up with a wrong answer

Sent from my iPhone using GMAT Club Forum mobile app

you are so right, thanks! but I guess there must be a short cut to consider all such numbers, because its really time consuming... and I can`t imagine how it is doable in two minutes... there is a high risk that one number will be missing
Senior CR Moderator
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Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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16 Aug 2017, 01:28
2
Bunuel wrote:
A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25
(B) 2/9
(C) 2/15
(D) 1/9
(E) 1/10

From 100 to 999, there are 999 - 100 + 1 = 900 three-digit numbers,

From 10 unit digits 0, 1, 2, ..., 9, select 3 units digits. There are totally $$10C3=\frac{10!}{3!7!}=\frac{8*9*10}{2*3}=4*3*10=120$$

From each selection of 3 units digits, we could create a “descending number”. Hence, there are totally 120 “descending number”.

The probability is: $$\frac{120}{900}=\frac{2}{15}$$

_________________
Director
Joined: 09 Mar 2016
Posts: 648
Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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16 Aug 2017, 01:36
broall wrote:
Bunuel wrote:
A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25
(B) 2/9
(C) 2/15
(D) 1/9
(E) 1/10

From 100 to 999, there are 999 - 100 + 1 = 900 three-digit numbers,

From 10 unit digits 0, 1, 2, ..., 9, select 3 units digits. There are totally $$10C3=\frac{10!}{3!7!}=\frac{8*9*10}{2*3}=4*3*10=120$$

From each selection of 3 units digits, we could create a “descending number”. Hence, there are totally 120 “descending number”.

The probability is: $$\frac{120}{900}=\frac{2}{15}$$

hello ! I knew it , there was a shortcut method could you please break down it in more details

From 10 unit digits 0, 1, 2, ..., 9, select 3 units digits. There are totally $$10C3=\frac{10!}{3!7!}=\frac{8*9*10}{2*3}=4*3*10=120$$

I have basic knowledge of combinations and permutations, but I cant understand how did you apply it here
Intern
Joined: 28 May 2017
Posts: 18
Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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16 Aug 2017, 01:38
1
2
In selection of 10digits, where we can choose 3 digits. Also remember in set of 3digits there is only one possible set which can be in ascending or descending order. Example for a set {1,2,3} we have 123, 132, 213, 231, 312, 321.
So for descending we have a choice C(10,3) = 120

So 120/900 = 2/15

Sent from my iPhone using GMAT Club Forum
Senior CR Moderator
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1393
Location: Viet Nam
Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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16 Aug 2017, 01:39
dave13 wrote:
hello ! I knew it , there was a shortcut method could you please break down it in more details

From 10 unit digits 0, 1, 2, ..., 9, select 3 units digits. There are totally $$10C3=\frac{10!}{3!7!}=\frac{8*9*10}{2*3}=4*3*10=120$$

I have basic knowledge of combinations and permutations, but I cant understand how did you apply it here

Cant get your ideas. Could you specify more?
_________________
Intern
Joined: 28 May 2017
Posts: 18
Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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16 Aug 2017, 01:41
I am sending you a direct message of the explanation

Sent from my iPhone using GMAT Club Forum
Intern
Joined: 23 Jun 2017
Posts: 4
GPA: 4
WE: Information Technology (Computer Software)
Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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17 Aug 2017, 06:02
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Soln.png [ 35.23 KiB | Viewed 1482 times ]

Please refer to the attached image for detailed solution .
Intern
Joined: 12 Sep 2017
Posts: 28
Re: A “descending number” is a three-digit number, such that the units dig [#permalink]

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08 Jun 2018, 11:20
Bunuel wrote:
A “descending number” is a three-digit number, such that the units digit is less than the tens digits and the tens digit is less than the hundreds digit. What is the probability that a three-digit number chosen at random is a “descending number”?

(A) 3/25

(B) 2/9

(C) 2/15

(D) 1/9

(E) 1/10

Hi
My approach is -

list of numbers to make a 3 digit number in descending order - 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 (total 10 numbers).
x > y > z
so x can be filled with 8 possible number because we need 2 number y and z after that. hence, total number of ways in which a descending number can be made is 8 x 2 x 1.
and total number of ways to make 3 digit numbers are 10C3 (120)

8 x 2 x 1 / 120 = 16 / 120
2/15
Re: A “descending number” is a three-digit number, such that the units dig   [#permalink] 08 Jun 2018, 11:20
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