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talaangoshtari: Master | Next Rank: 500 Posts; Posts: 154; Joined: Wed May 21, 2014 4:29 amFollowed by: 1 member; Thanked: 8 times

by talaangoshtari » Sat May 16, 2015 8:44 am

How many positive integers less than 10,000 are such that the sum of their digits is 5?

A.56
B.22
C.100
D.75
E.80

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Source: Beat The GMAT — Problem Solving | Sat May 16, 2015 8:44 am

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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 » Sat May 16, 2015 8:49 am

talaangoshtari wrote:How many positive integers less than 10,000 are such that the sum of their digits is 5?

(a) 31
(b) 51
(c) 56
(d) 62
(e) 93

NOTE: I have replaced talaangoshtari's answer choices with the answer choices that appeared in the original question.

The answer is C

More on this question here: https://www.beatthegmat.com/very-tricky- ... 25349.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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MartyMurray: Legendary Member; Posts: 2135; Joined: Mon Feb 03, 2014 9:26 amFollowed by: 140 members; Location: https://martymurraycoaching.com/; Thanked: 955 times; GMAT Score:800

by MartyMurray » Fri May 22, 2015 9:30 pm

talaangoshtari wrote:How many positive integers less than 10,000 are such that the sum of their digits is 5?

(a) 31
(b) 51
(c) 56
(d) 62
(e) 93

Positive numbers less than 10,000 can have one to four digits.

One digit numbers whose digits add up to 5 can only use the number 5.

5

So there is just 1 one digit number whose digits add up to 5.

Two digit numbers whose digits add up to 5 can be permutations of (1,4), (2,3), and (5,0).

Permutations of two elements have two arrangements, except for (5,0) which can only have one, 50, as 05 is not really a two digit number.

So 2 + 2 + 1 = 5 two digit numbers.

Three digit numbers whose digits add up to five can be permutations of (1,1,3), (0,1,4), (0,2,3), (0,0,5), and (2,2,1).

For example (1,1,3) can be arranged 3!/2! = 3 ways.

However, because of the zeros, (0,0,5) can only be arranged one way, as 500.

Taking into account the doubles and the zeros, one can arrange those sets in the following number of ways respectively.

3 + 4 + 4 + 1 + 3 = 15 three digit numbers.

Four digit numbers whose digits add up to five can be permutations of (1,1,3,0), (0,0,1,4), (0,0,2,3), (0,0,0,5), (0,2,2,1) and (1,1,1,2).

For example, (1,1,3,0) can be arranged (3 x 3 x 2 x 1)/2! = 9 ways.

(1,1,1,2) can be arranged 4!/3! = 4 ways.

Taking into the account the doubles, triples and zeros, one can arrange those sets in the following number of ways respectively.

9 + 6 + 6 + 1 + 9 + 4 = 35

So the total of one digit, two digit, three digit, and four digit numbers is the following.

1 + 5 + 15 + 35 = 56

Choose C.

Marty Murray
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Mon May 25, 2015 1:22 am

Very Good Question, What is the source of such questions.

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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 » Mon May 25, 2015 7:04 am

I "created" this question in 2008 (which was based on another resource I had created earlier): https://www.beatthegmat.com/very-tricky- ... 25349.html

I had seen a similar question in one of my combinatorics classes in university.

This question is skirting the acceptable limits of the GMAT. Probably 800+

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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talaangoshtari: Master | Next Rank: 500 Posts; Posts: 154; Joined: Wed May 21, 2014 4:29 amFollowed by: 1 member; Thanked: 8 times

by talaangoshtari » Mon May 25, 2015 6:19 pm

I found this problem in my high school book.

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Mon May 25, 2015 7:53 pm

Good question, keep posting such questions

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