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Tough 700 level question

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by Anurag@Gurome » Wed Oct 05, 2011 7:08 pm
gvosough wrote:how many positive integers less than 10000 are there in which the sum of the digits equals 5?

Answer :56
Thanks
Nice question.
We need to find integers between 0 to 9999, in which the sum of digits adds up to 5.
(1) One digit is 5 and all other are 0: 0005, 0050, 0500, 5000 or we can say that no. of ways we can arrange the digits = 4!/3! = 4 ways
(2)Three 1's and one 2: 1112, this can be done in 4!/3! = 4 ways
(3) One 4 and one 1: 4100, this can be done in 4!/2! = 12 ways
(4) One 3 and one 2: 3200, this can be done in 4!/2! = 12 ways
(5) One 3 and two 1's: 3110, this can be done in 4!/2! = 12 ways
(6) Two 2's and One 1: 2210, this can be done in 4!/2! = 12 ways

Therefore, required number of positive integers = (12 * 4) + (4 * 2) = 48 + 8 = 56
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by GMATGuruNY » Wed Oct 05, 2011 7:55 pm
I posted an alternate solution here:

https://www.beatthegmat.com/experts-any- ... 82307.html
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by Cheese12 » Thu Oct 06, 2011 8:13 am
GMATGuruNY wrote:I posted an alternate solution here:

https://www.beatthegmat.com/experts-any- ... 82307.html
Hi,

Thanks for the great solution!... But i have a lil doubt regarding the first case you listed down

Thus:
"|||DDDDD = 5. " ->>> if we consider 5 digits say 50,000.. isnt this larger than 10,000 ?
||DDDD|D = 41.
|D|DDD|D = 131.
D|D|D|DD = 1112.

Kindly help clarify.. thanks !! :)
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by GMATGuruNY » Thu Oct 06, 2011 8:28 am
Cheese12 wrote:
GMATGuruNY wrote:I posted an alternate solution here:

https://www.beatthegmat.com/experts-any- ... 82307.html
Hi,

Thanks for the great solution!... But i have a lil doubt regarding the first case you listed down

Thus:
"|||DDDDD = 5. " ->>> if we consider 5 digits say 50,000.. isnt this larger than 10,000 ?
||DDDD|D = 41.
|D|DDD|D = 131.
D|D|D|DD = 1112.

Kindly help clarify.. thanks !! :)
|||DDDDD = 0005.

The five D's represent not 5 digits but the 5-UNIT SUM that the four digits must add up to.

To avoid confusion, perhaps U would be a better letter than D to stand in for the 5-unit sum:

|||UUUUU = 0005.
UUUUU||| = 5000.
||UU|UUU = 0023.
etc.
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by Cheese12 » Thu Oct 06, 2011 8:52 am
GMATGuruNY wrote:
Cheese12 wrote:
GMATGuruNY wrote:I posted an alternate solution here:

https://www.beatthegmat.com/experts-any- ... 82307.html
Hi,

Thanks for the great solution!... But i have a lil doubt regarding the first case you listed down

Thus:
"|||DDDDD = 5. " ->>> if we consider 5 digits say 50,000.. isnt this larger than 10,000 ?
||DDDD|D = 41.
|D|DDD|D = 131.
D|D|D|DD = 1112.

Kindly help clarify.. thanks !! :)
|||DDDDD = 0005.

The five D's represent not 5 digits but the 5-UNIT SUM that the four digits must add up to.

To avoid confusion, perhaps U would be a better letter than D to stand in for the 5-unit sum:

|||UUUUU = 0005.
UUUUU||| = 5000.
||UU|UUU = 0023.
etc.

Ok got it now :) ... thanks!!
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