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# How many integers n between 100 and 299 are there such that the hundre

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Math Expert
Joined: 02 Sep 2009
Posts: 51215
How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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26 Nov 2018, 23:33
00:00

Difficulty:

55% (hard)

Question Stats:

68% (01:43) correct 32% (02:11) wrong based on 72 sessions

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How many integers n between 100 and 299 are there such that the hundreds digit of n is less than the tens digit of n and the tens digit of n is less than the units digit of n ?

A. 48
B. 49
C. 50
D. 51
E. 52

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Director
Joined: 18 Jul 2018
Posts: 501
Location: India
Concentration: Finance, Marketing
WE: Engineering (Energy and Utilities)
Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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27 Nov 2018, 01:30
1
First number N such that the hundreds digit of n is less than the tens digit of n and the tens digit of n is less than the units digit of n is 123.
Then 124 ,125 ,126, 127, 128, 129 - Total 7 numbers.
Next is 134-139 - Total 6 numbers.
145-149 - 5 numbers.
156-159 - 4 numbers.
167-169 - 3 numbers.
178-179 - 2 numbers.
180 - 1 number.
total of 28 numbers between 100-200.

Similarly from 200 to 299.
234-239 - 6 numbers.
245-249 - 5 numbers.
256-259 - 4 numbers.
267-269 - 3 numbers.
278-279 - 2 numbers.
289 - 1 number.
total of 21 numbers.

Total is 28+21 = 49 numbers.

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Director
Joined: 18 Aug 2017
Posts: 742
Location: India
Concentration: Sustainability, Marketing
WE: Marketing (Energy and Utilities)
Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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27 Nov 2018, 09:53
Bunuel wrote:
How many integers n between 100 and 299 are there such that the hundreds digit of n is less than the tens digit of n and the tens digit of n is less than the units digit of n ?

A. 48
B. 49
C. 50
D. 51
E. 52

GMATinsight :
Sir is there any other method to solve such question, where we dont end up counting every number?

N : (100A+10B+C)
so we need to find such specific no. ie 100A<10B<C

123,124,125,126,127,128,129,134,135,136.... so on..

this is a very time consuming method is there any short cut??
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VP
Joined: 07 Dec 2014
Posts: 1128
Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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28 Nov 2018, 15:14
Bunuel wrote:
How many integers n between 100 and 299 are there such that the hundreds digit of n is less than the tens digit of n and the tens digit of n is less than the units digit of n ?

A. 48
B. 49
C. 50
D. 51
E. 52

this is a triangular number sequence:
0,1,3,6,10,15,21,28...
here is the relationship between hundreds unit and number of possible integers:
8-0
7-1
6-3
5-6
4-10
3-15
2-21
1-28
because the value for term n=n(n-1)/2,
then the 7th term, 2, will have 7*6/2=21 possible integers
and the 8th term, 1, will have 8*7/2=28 possible integers
21+28=49 total integers
B
Intern
Joined: 10 Oct 2017
Posts: 3
Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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29 Nov 2018, 23:25
If the hundreds digit is 1, the tens digit has 7 cases, the units digit has 7 cases => has total 28 numbers
If the hundreds digit is 2, the tens digit has 6 cases, the units digit has 6 cases => has total 21 numbers
--> has 49 numbers
--> B
Manager
Joined: 08 Jul 2018
Posts: 60
Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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29 Nov 2018, 23:31
duynguyen612 wrote:
If the hundreds digit is 1, the tens digit has 7 cases, the units digit has 7 cases => has total 28 numbers
If the hundreds digit is 2, the tens digit has 6 cases, the units digit has 6 cases => has total 21 numbers
--> has 49 numbers
--> B

Can you please explain how you get 28 and 21 numbers?
Manager
Joined: 17 May 2015
Posts: 249
Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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30 Nov 2018, 02:00
Bunuel wrote:
How many integers n between 100 and 299 are there such that the hundreds digit of n is less than the tens digit of n and the tens digit of n is less than the units digit of n ?

A. 48
B. 49
C. 50
D. 51
E. 52

Hi,

We can divide the whole range into the two parts: 100 to 199 and 200 to 299.

100 to 199

Hundreds digit is fixed in this case,i.e. 1. Tens digit can be 2,3,4,5,6,7, and 8.
For example, when tens digit is 2 we have the following possibilities:
Attachment:

Number_counting1.png [ 4.05 KiB | Viewed 249 times ]

Number of ways = 1*1*7 = 7 ways.

Similarly, when tens digit is 3 we have the following possibilities:

Attachment:

Number_counting2.png [ 3.9 KiB | Viewed 249 times ]

Number of ways = 1*1*6 = 6 ways.

Hence, the total number of ways = $$7 + 6 + \dots + 1 = \frac{7\times8}{2} = 28$$ numbers.

200 to 199

Hundreds number is 2. Tens digit can be 3,4,5,6,7, and 8. Hundreds digit can be 4,5,6,7,8, and 9.

Hence, the total number in this case = $$6 + 5 + \dots + 1 = \frac{6\times7}{2} = 21$$ numbers.

Total number = 28 + 21 = 49 numbers.

Thanks.
Re: How many integers n between 100 and 299 are there such that the hundre &nbs [#permalink] 30 Nov 2018, 02:00
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