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

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Joined: 31 Oct 2012
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How many positive integers less than 10,000 [#permalink]

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31 Oct 2012, 15:42
Hi everybody
this is my first topic and i am really looking for your help

This is an old Q
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

[Reveal] Spoiler:
and the Answer is (C) 56

using the "stars and bars method " or "separator method".

but when using the same approach with different numbers (bigger than 1o) => this method doesn't work

e.g.
lets find out 35 or 36 instead of 5

How many positive integers less than 10,000 are there in which the sum of the digits equals 36?

the only correct answer is one integer which is 9999 => 9+9+9+9
but when using that method
39!/(36! 3!) = 9139 integers instead of 1

and the same 35
correct answer is 4 ( 9998, 9989 , 9899, 8999 )
but using the method
38!/(35! 3!) = 8436

really confusing
any help

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Re: How many positive integers less than 10,000 [#permalink]

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31 Oct 2012, 21:15
ememem wrote:
Hi everybody
this is my first topic and i am really looking for your help

This is an old Q
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

[Reveal] Spoiler:
and the Answer is (C) 56

using the "stars and bars method " or "separator method".

but when using the same approach with different numbers (bigger than 1o) => this method doesn't work

e.g.
lets find out 35 or 36 instead of 5

How many positive integers less than 10,000 are there in which the sum of the digits equals 36?

the only correct answer is one integer which is 9999 => 9+9+9+9
but when using that method
39!/(36! 3!) = 9139 integers instead of 1

and the same 35
correct answer is 4 ( 9998, 9989 , 9899, 8999 )
but using the method
38!/(35! 3!) = 8436

really confusing
any help

The method works for 5 because each grouping of ones in this case is lesser than 10.

ie, each grouping can be equated to a single digit. So the highest number that would work for this method can be only 9.

Kudos Please... If my post helped.
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Re: Integers less than 10,000 [#permalink]

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25 May 2013, 22:03
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 $$\frac{8!}{5!3!}=56$$.

Answer: C (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: $${n+r-1}_C_{r-1}={5+4-1}_C_{4-1}={8}C3=\frac{8!}{5!3!}=56$$

Also see the image I found in the net about this question explaining the concept:
Attachment:
pTNfS-2e270de4ca223ec2741fa10b386c7bfe.jpg

Hi Bunnel,

Can such questions come up on gmat ?

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Re: Integers less than 10,000 [#permalink]

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26 May 2013, 03:45
Bunuel wrote:
anilnandyala wrote:
thanks Bunuel
can u explain me this by using the formulae
How many positive integers less than 10,000 are there in which the sum of the digits equals 6?
thanks in advance

6 * (digits) and 3 ||| --> ******||| --> # of permutations of these symbols is $$\frac{9!}{6!3!}$$.

Or: The total number of ways of dividing n identical items (6 *'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 6 in our case) is $${n+r-1}_C_{r-1}$$.

In our case we'll get: $${n+r-1}_C_{r-1}={6+4-1}_C_{4-1}={9}C3=\frac{9!}{6!3!}$$.

Hope it's clear.

Hi Bunnel,

Can I say that this involves the placement of 5 identical 1's in four places such that each place can receive 0 to 5 1's.

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Re: Integers less than 10,000 [#permalink]

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26 May 2013, 04:47
cumulonimbus wrote:
Bunuel wrote:
anilnandyala wrote:
thanks Bunuel
can u explain me this by using the formulae
How many positive integers less than 10,000 are there in which the sum of the digits equals 6?
thanks in advance

6 * (digits) and 3 ||| --> ******||| --> # of permutations of these symbols is $$\frac{9!}{6!3!}$$.

Or: The total number of ways of dividing n identical items (6 *'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 6 in our case) is $${n+r-1}_C_{r-1}$$.

In our case we'll get: $${n+r-1}_C_{r-1}={6+4-1}_C_{4-1}={9}C3=\frac{9!}{6!3!}$$.

Hope it's clear.

Hi Bunnel,

Can I say that this involves the placement of 5 identical 1's in four places such that each place can receive 0 to 5 1's.

Yes, that's correct.
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Re: How many positive integers less than 10,000 are there in [#permalink]

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05 Jul 2013, 02:27
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: How many positive integers less than 10,000 are there in [#permalink]

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12 Jul 2013, 22:44
alphabeta1234 wrote:
Bunuel,

Many apologies for reviving this method. But I gotta say I love your "stars and bars" method. My Question is it seems to only work for this question if the sum is less than 9 for this question. You can't have more than 9 stars in each slot. What if the question had asked,

How many positive integers less than 10,000 are there in which the sum of the digits equals 13?

Now this is a little tricker. Here is another example from another problem posted on this forum that illustrates the problem. Is there a way we can exand "stars and bars" to numbers that sum larger than 9?

A wheel of fortune contains numerical values from 1 to 8. The scoring system of the game is based on the sum of these values. If the host were to spin the wheel three times, how many possible number of combinations are there that will give the player the sum of 16 points?

Bunuel,

Like alphabeta1234 mentioned above, there is a limitation to the stars and bars method. How then would we be able to find out the number of positive integers less than 10,000 if their sum is to be, say, 13? He mentioned the wheel of fortune method, but looking it up on the internet didn't turn up much that was useful.

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Re: How many positive integers less than 10,000 are there in [#permalink]

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13 Jul 2013, 00:10
keylimepie wrote:
alphabeta1234 wrote:
Bunuel,

Many apologies for reviving this method. But I gotta say I love your "stars and bars" method. My Question is it seems to only work for this question if the sum is less than 9 for this question. You can't have more than 9 stars in each slot. What if the question had asked,

How many positive integers less than 10,000 are there in which the sum of the digits equals 13?

Now this is a little tricker. Here is another example from another problem posted on this forum that illustrates the problem. Is there a way we can exand "stars and bars" to numbers that sum larger than 9?

A wheel of fortune contains numerical values from 1 to 8. The scoring system of the game is based on the sum of these values. If the host were to spin the wheel three times, how many possible number of combinations are there that will give the player the sum of 16 points?

Bunuel,

Like alphabeta1234 mentioned above, there is a limitation to the stars and bars method. How then would we be able to find out the number of positive integers less than 10,000 if their sum is to be, say, 13? He mentioned the wheel of fortune method, but looking it up on the internet didn't turn up much that was useful.

There is a direct formula given on previous page.
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Re: How many positive integers less than 10,000 are there in [#permalink]

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13 Jul 2013, 00:52
Bunuel wrote:
keylimepie wrote:
alphabeta1234 wrote:
Bunuel,

Many apologies for reviving this method. But I gotta say I love your "stars and bars" method. My Question is it seems to only work for this question if the sum is less than 9 for this question. You can't have more than 9 stars in each slot. What if the question had asked,

How many positive integers less than 10,000 are there in which the sum of the digits equals 13?

Now this is a little tricker. Here is another example from another problem posted on this forum that illustrates the problem. Is there a way we can exand "stars and bars" to numbers that sum larger than 9?

A wheel of fortune contains numerical values from 1 to 8. The scoring system of the game is based on the sum of these values. If the host were to spin the wheel three times, how many possible number of combinations are there that will give the player the sum of 16 points?

Bunuel,

Like alphabeta1234 mentioned above, there is a limitation to the stars and bars method. How then would we be able to find out the number of positive integers less than 10,000 if their sum is to be, say, 13? He mentioned the wheel of fortune method, but looking it up on the internet didn't turn up much that was useful.

There is a direct formula given on previous page.

I'm sorry. I'm afraid I don't see which one I can apply to the situation. Could you please point me in the right direction?

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Re: How many positive integers less than 10,000 are there in [#permalink]

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13 Jul 2013, 00:58
keylimepie wrote:
There is a direct formula given on previous page.

I'm sorry. I'm afraid I don't see which one I can apply to the situation. Could you please point me in the right direction?[/quote]

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

For example, here: how-many-positive-integers-less-than-10-000-are-there-in-85291.html#p710836

Notice that it's highly unlikely you'll need it for the GMAT.
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Re: How many positive integers less than 10,000 are there in [#permalink]

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13 Jul 2013, 01:35
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]

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25 Nov 2013, 05:18
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]

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29 Nov 2013, 11:03
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]

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29 Nov 2013, 11:09
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]

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27 Dec 2013, 15:33
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]

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05 May 2014, 17:39
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]

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30 Nov 2014, 14:45
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]

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15 Dec 2014, 05:33
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 $$\frac{8!}{5!3!}=56$$.

Answer: C (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: $${n+r-1}_C_{r-1}={5+4-1}_C_{4-1}={8}C3=\frac{8!}{5!3!}=56$$

Also see the image I found in the net about this question explaining the concept:
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]

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26 Dec 2014, 04:43
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
Adding them up: 9+18+27= 54.

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,
Natalia

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Re: How many positive integers less than 10,000 are there in [#permalink]

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18 Jan 2015, 07:52
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] 18 Jan 2015, 07:52

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