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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: 58452
How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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27 Nov 2018, 00:33
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65% (hard)

Question Stats:

59% (02:32) correct 41% (02:21) wrong based on 117 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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Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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

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27 Nov 2018, 10:53
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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How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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Updated on: 16 Dec 2018, 20:53
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

this is a triangular number sequence:
0,1,3,6,10,15,21,28...
here is the relationship between the hundreds block and number of compliant integers in each:
800-0
700-1
600-3
500-6
400-10
300-15
200-21
100-28
because the value for term n=n(n-1)/2,
the 7th term (200 block) will have 7*6/2=21 compliant integers
and the 8th term (100 block) will have 8*7/2=28 compliant integers
21+28=49 total compliant integers between 100 and 299
B

Originally posted by gracie on 28 Nov 2018, 16:14.
Last edited by gracie on 16 Dec 2018, 20:53, edited 1 time in total.
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Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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30 Nov 2018, 00:25
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
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Posts: 63
Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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30 Nov 2018, 00:31
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?
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Joined: 17 May 2015
Posts: 247
Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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30 Nov 2018, 03:00
1
1
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 2166 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 2160 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.
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Re: How many integers n between 100 and 299 are there such that the hundre  [#permalink]

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06 Jan 2019, 12:50
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

Solving using counting principle:

abc is a number in between 100 and 299, a is hundreds, b is tens and c is units place.
0<a<b<c<9
So, c is atleast 3.

✓c can take values between 3-9, inclusive = 7 values.
AND
✓b can take values between 2 and 8, inclusive (b is atleast 2) = 7 values.
AND
✓a can take 1 value = 1

Total possibilities = 7x7x1 = 49

Posted from my mobile device
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Re: How many integers n between 100 and 299 are there such that the hundre   [#permalink] 06 Jan 2019, 12:50
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