sum of the digits equals 5

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Thu Feb 24, 2011 4:02 am

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

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Source: Beat The GMAT — Problem Solving | Thu Feb 24, 2011 4:02 am

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Thu Feb 24, 2011 4:33 am

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

The correct answer is C.

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anshumishra: Legendary Member; Posts: 543; Joined: Tue Jun 15, 2010 7:01 pmFollowed by: 3 members; Thanked: 147 times

by anshumishra » Thu Feb 24, 2011 5:15 am

sanju09 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

Source: https://readyforgmat.com

Think about the question in this way: a total of 5 to be distributed among the 4 digits. we have to find the number of ways it can be distributed.

Let * represent a sum of 1, and | represent a seperator between two digits. So,we have 5 * and 3 |.

Some of the ways it can be represented as :
*|**|**| -> 1220
***|||** -> 3002
and so on.

So the number of ways to select 3 | out of a total of 8 (5 * and 3 |) = 8C3 = 56.
C

Thanks
Anshu

(Every mistake is a lesson learned )

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