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

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rakeshd347: Master | Next Rank: 500 Posts; Posts: 391; Joined: Sat Mar 02, 2013 5:13 amFollowed by: 4 members; Thanked: 50 times

by rakeshd347 » Sun Oct 06, 2013 8:06 pm

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

OA is C

I have a confusion about this question. When we are solving with the separator method. We are considering 5 numbers to be distributed among 4 people. So we have 5 numbers and 3 separators. This way will give 8!/5!3!....but my question is aren't we considering 0,1,2,3,4,5 so we have 6 digits to distribute to 4 people. Can you please spot where I am wrong. I know We don't have to consider 0 and only 1,2,3,4,5....but 5000 is also the digit which have sum equal to 5.

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Source: Beat The GMAT — Problem Solving | Sun Oct 06, 2013 8:06 pm

vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pmFollowed by: 4 members; Thanked: 29 times

by vipulgoyal » Sun Oct 06, 2013 8:33 pm

I have come across with brute force method, which took me 10 minuts and impractial in exam scenarios

one digit No - 5 -- 1
two digit no - 14,41,23,32,50 -- 5
three digit - 113(3 ways) , 221(3 ways), 500(one way), 401(4 ways), 302(4 ways) -- 15
four digit - 1112(4 ways), 1220(9 ways),1310(9ways),1004(6ways),2003(6 ways), 5000 (one way) -- 35 ways

35+15+5+1= 56

I didnt get what you are trying to ask??

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Sun Oct 06, 2013 8:40 pm

Digits : _ _ _ _

We need to distribute : 1 1 1 1 1 = 5

Using Seperator Method:

_ | _ | _ | _

Hence, 8!/5!3! = 8*7 = 56

Answer [spoiler]{C}[/spoiler]

R A H U L

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rakeshd347: Master | Next Rank: 500 Posts; Posts: 391; Joined: Sat Mar 02, 2013 5:13 amFollowed by: 4 members; Thanked: 50 times

by rakeshd347 » Sun Oct 06, 2013 8:58 pm

theCodeToGMAT wrote:Digits : _ _ _ _

We need to distribute : 1 1 1 1 1 = 5

Using Seperator Method:

_ | _ | _ | _

Hence, 8!/5!3! = 8*7 = 56

Answer [spoiler]{C}[/spoiler]

Hi Rahul,

This is where I am getting confused. You have used four _ and three separators I so the total number is 7. So shouldn't it be 7!/4!3!

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sun Oct 06, 2013 9:07 pm

rakeshd347 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

My solution can be found here: https://www.beatthegmat.com/very-tricky- ... 25349.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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rakeshd347: Master | Next Rank: 500 Posts; Posts: 391; Joined: Sat Mar 02, 2013 5:13 amFollowed by: 4 members; Thanked: 50 times

by rakeshd347 » Sun Oct 06, 2013 9:09 pm

theCodeToGMAT wrote:Digits : _ _ _ _

We need to distribute : 1 1 1 1 1 = 5

Using Seperator Method:

_ | _ | _ | _

Hence, 8!/5!3! = 8*7 = 56

Answer [spoiler]{C}[/spoiler]

Rahul I think I have got the answer.

5 Different chocolate has to distributed among 4 kids....this question is same as your question as we are considering 4 digit numbers.
so we need 3 separator.
OOOOO and III
if its 0005 then it will represent....III00000
if its 1211 then it will represent....0I00I00

So it counts to 8 so the answer is 8!/3!5!

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Sun Oct 06, 2013 9:19 pm

rakeshd347 wrote:
theCodeToGMAT wrote:Digits : _ _ _ _

We need to distribute : 1 1 1 1 1 = 5

Using Seperator Method:

_ | _ | _ | _

Hence, 8!/5!3! = 8*7 = 56
Answer [spoiler]{C}[/spoiler]
Rahul I think I have got the answer.

5 Different chocolate has to distributed among 4 kids....this question is same as your question as we are considering 4 digit numbers.
so we need 3 separator.
OOOOO and III
if its 0005 then it will represent....III00000
if its 1211 then it will represent....0I00I00

So it counts to 8 so the answer is 8!/3!5!

Bingo!

R A H U L

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