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# How many positive integers less than 10,000 are there in

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Re: How many positive integers less than 10,000 are there in [#permalink]  13 Jul 2013, 00:35
Bunuel wrote:

It's everywhere: on the 1st page, on the 2nd...

Notice that it's highly unlikely you'll need it for the GMAT.

I understood the various explanations there. But when employing that, isn't there the worry that you might have

XXXXXXXXXXXXX|||| ?
(There are 13 X's there.)

Since numbers in decimal only go up to 9, wouldn't the cases where there are 10 or more X's grouped together be a problem?
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Re: How many positive integers less than 10,000 are there in [#permalink]  25 Nov 2013, 04:18
Excellent solution Bunuel. Saves a minute at minimum !!!
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Re: How many positive integers less than 10,000 are there in [#permalink]  29 Nov 2013, 10:03
Hi Bunnel, please can explain when the separator concept is to be used and how to use it. Basically i did not understand in this question that why have we considered only 4 digit number. Please help.
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Re: How many positive integers less than 10,000 are there in [#permalink]  29 Nov 2013, 10:09
Expert's post
nayan19 wrote:
Hi Bunnel, please can explain when the separator concept is to be used and how to use it. Basically i did not understand in this question that why have we considered only 4 digit number. Please help.

Integers less than 10,000 are 1, 2, or 3-digit numbers. Post here: how-many-positive-integers-less-than-10-000-are-there-in-85291.html#p710836 explains that we can get single-digit as well as 2 or 3-digit numbers with that approach (check the examples there).

Similar questions to practice:
larry-michael-and-doug-have-five-donuts-to-share-if-any-108739.html
in-how-many-ways-can-5-different-rings-be-worn-in-four-126991.html

Hope this helps.
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Re: How many positive integers less than 10,000 are there in [#permalink]  27 Dec 2013, 14:33
Exceptionnal technique! Thanks all for this! Saves a lot of time in a lot of situations!

Incredible minds!

thanks !!
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Re: How many positive integers less than 10,000 are there in [#permalink]  05 May 2014, 16:39
Bunuel's method was clearly simpler and faster, though I would hardly come up with a similar solution in the gmat.

I did it in a different way, can someone check if the approach was valid?

5 and 0s: 4P1*3C3 = 4*1 = 4
4, 1 and 0s: 4P1*3P1*2C2 = 4*3*1 = 12
3, 2 and 0s: 4P1*3P1*2C2 = 4*3*1 = 12
2, 2, 1 and 0: 4P2*2P1*1 = 12*2 = 24
2, 1, 1, 1: 4P1*3C3 = 4*1 = 4

4 + 12+ 12 + 24 + 4 = 56
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Re: How many positive integers less than 10,000 are there in [#permalink]  30 Nov 2014, 13:45
Hey Bunel
would you plz tell me in this formula 8! / (5!*3!) ,you got the numbers 8,5, and 3 from where ???
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How many positive integers less than 10,000 are there in [#permalink]  15 Dec 2014, 04:33
Bunuel wrote:
Ramsay wrote:
Sorry guys,

Could someone please explain the following:

"There are 8C3 ways to determine where to place the separators"

I'm not familiar with this shortcut/approach.

Ta

Consider this: we have 5 $$d$$'s and 3 separators $$|$$, like: $$ddddd|||$$. How many permutations (arrangements) of these symbols are possible? Total of 8 symbols (5+3=8), out of which 5 $$d$$'s and 3 $$|$$'s are identical, so $$\frac{8!}{5!3!}=56$$.

With these permutations we'll get combinations like: $$|dd|d|dd$$ this would be 3 digit number 212 OR $$|||ddddd$$ this would be single digit number 5 (smallest number less than 10,000 in which sum of digits equals 5) OR $$ddddd|||$$ this would be 4 digit number 5,000 (largest number less than 10,000 in which sum of digits equals 5)...

Basically this arrangements will give us all numbers less than 10,000 in which sum of the digits (sum of 5 d's=5) equals 5.

Hence the answer is $$\frac{8!}{5!3!}=56$$.

This can be done with direct formula as well:

The total number of ways of dividing n identical items (5 d's in our case) among r persons or objects (4 digt places in our case), each one of whom, can receive 0, 1, 2 or more items (from zero to 5 in our case) is $${n+r-1}_C_{r-1}$$.

In our case we'll get: $${n+r-1}_C_{r-1}={5+4-1}_C_{4-1}={8}C3=\frac{8!}{5!3!}=56$$

Attachment:
pTNfS-2e270de4ca223ec2741fa10b386c7bfe.jpg

Hi Bunuel,

Could you please clarify why we are taking 5 d's and 3 seprator (/). i am getting confusion here. we can take four separator also and get the result.

Thanks.
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Re: How many positive integers less than 10,000 are there in [#permalink]  26 Dec 2014, 03:43
Hi,

If I may venture to propose the solution I used and you can tell me what I am missing.

I started by testing it from 0-10. There is one such number (5). From 11-20 there is one such number (14). This led me realize than from 0-99 there are 9 such numbers. So, this was the lengthy part of my thinking process (not lengthy at all).

From 0-99: 9 numbers.
From : 100-999: 9*2= 18 numbers
From 1000-9999: 9*3= 27 numbers

This is close enough so I decided to choose 56 anyway, but since there are 2 numbers missing, could you tell me why and where?

Thank you,
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Re: How many positive integers less than 10,000 are there in [#permalink]  18 Jan 2015, 06:52
AKProdigy87 wrote:
I believe the answer to be C: 56.

Basically, the question asks how many 4 digit numbers (including those in the form 0XXX, 00XX, and 000X) have digits which add up to 5. Think about the question this way: we know that there is a total of 5 to be spread among the 4 digits, we just have to determine the number of ways it can be spread.

Let X represent a sum of 1, and | represent a seperator between two digits. As a result, we will have 5 X's (digits add up to the 5), and 3 |'s (3 digit seperators).

So, for example:

XX|X|X|X = 2111
||XXX|XX = 0032

etc.

There are 8C3 ways to determine where to place the separators. Hence, the answer is 8C3 = 56.

Sorry i could not understand how you have started with 3 Separators , why not 4 or 5 ?
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Re: How many positive integers less than 10,000 are there in [#permalink]  16 Jun 2015, 22:59
Bunuel wrote:
How many positive integers less than 10,000 are there in which the sum of the digits equals 5?

(A) 31
(B) 51
(C) 56
(D) 62
(E) 93

from 0 to 9, 5 ( 1 number)
from 10 to 99, we have 14,23,32,41,50 (5 numbers)
from 100 to 999, we have 104,113. 122,131, 140, 203,212,221, 230,302, 311,320,401,410,500 (15 numbers)
from 1000 to 9999, we have 1004,1040,1103,1112, 1121,1130,1202,1211,1220,1301,1310,1400, 2003,2012,2021,2030,2102,2111,2120,2201,2210,2300,3002.3011.3020,3101,3110,3200,4001,4010,4100,5000 (32 numbers)
So we have only 53 numbers.
Can anyone tell the numbers which I miss?
Re: How many positive integers less than 10,000 are there in   [#permalink] 16 Jun 2015, 22:59

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