Very tricky counting problem

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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 Dec 13, 2008 6:01 pm

This one is well past the 800-level, but there are a lot of clever people on this site who will, I imagine, accept the challenge.

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
Answer: C

Try to resist examining each case and counting all of the possibilities. To discourage this strategy I could have asked "How many positive integers less than 1,000,000 are there in which the sum of the digits equals 8?"

Source: Beat The GMAT — Problem Solving | Sat Dec 13, 2008 6:01 pm

cramya: Legendary Member; Posts: 2467; Joined: Thu Aug 28, 2008 6:14 pmFollowed by: 11 members; Thanked: 331 times

by cramya » Sat Dec 13, 2008 6:06 pm

Brent,
Thanks a bunch for providing the OA. By the way, welcome to this forum.

Please refer to
https://www.beatthegmat.com/need-help-wi ... 25206.html

Good luck with ur preps!

Regards,
Cramya

dmateer25: Community Manager; Posts: 1049; Joined: Sun Apr 06, 2008 5:15 pmFollowed by: 27 members; Location: Pittsburgh, PA; Thanked: 113 times; GMAT Score:710

by dmateer25 » Sat Dec 13, 2008 6:13 pm

Brent, did you create these 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 » Sat Dec 13, 2008 8:13 pm

Yes - I guess some former students have posted them here already.
You can find a downloadable set of advanced math questions at www.ReadyForGMAT.com

logitech: Legendary Member; Posts: 2134; Joined: Mon Oct 20, 2008 11:26 pmFollowed by: 25 members; Thanked: 237 times; GMAT Score:730

by logitech » Sat Dec 13, 2008 8:45 pm

Brent Hanneson wrote:Yes - I guess some former students have posted them here already.
You can find a downloadable set of advanced math questions at www.ReadyForGMAT.com

cool!

https://www.readyforgmat.com/math/docume ... rsion2.pdf

LGTCH
---------------------
"DON'T LET ANYONE STEAL YOUR DREAM!"

krazzy4: Newbie | Next Rank: 10 Posts; Posts: 9; Joined: Sat Apr 26, 2008 7:42 am

by krazzy4 » Sun Dec 14, 2008 12:37 am

Brent Hanneson wrote:Yes - I guess some former students have posted them here already.
You can find a downloadable set of advanced math questions at www.ReadyForGMAT.com

thanks for the link...cool resource.

dmateer25: Community Manager; Posts: 1049; Joined: Sun Apr 06, 2008 5:15 pmFollowed by: 27 members; Location: Pittsburgh, PA; Thanked: 113 times; GMAT Score:710

by dmateer25 » Sun Dec 14, 2008 5:25 am

Brent,

Thanks for taking the time to help out!

adilka: Senior | Next Rank: 100 Posts; Posts: 79; Joined: Thu Oct 23, 2008 9:28 am; Location: Canada; Thanked: 1 times; GMAT Score:700

by adilka » Sun Dec 14, 2008 1:46 pm

Thanks for the link, Brent

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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 Dec 14, 2008 3:02 pm

I checked the link to the previous solutions to this question, but didn't see any shortcuts, so I'll post this solution.

We can extend this solution and conclude that the number of integers less than 1,000,000 in which the sum of the digits equals 8 will be 13C5

mental: Master | Next Rank: 500 Posts; Posts: 145; Joined: Mon Sep 29, 2008 1:14 am; Thanked: 13 times

by mental » Sun Dec 14, 2008 11:09 pm

amazed

saketk: Legendary Member; Posts: 608; Joined: Sun Jun 19, 2011 11:16 amFollowed by: 8 members; Thanked: 37 times

by saketk » Sun Sep 18, 2011 3:27 am

Brent@GMATPrepNow wrote:This one is well past the 800-level, but there are a lot of clever people on this site who will, I imagine, accept the challenge.

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
Answer: C

Try to resist examining each case and counting all of the possibilities. To discourage this strategy I could have asked "How many positive integers less than 1,000,000 are there in which the sum of the digits equals 8?"

Hi Brent -- Got this link from your latest post..

This is how I did it (though I used bit of a counting here)

------------------------------------------------
We can make numbers from ---0,1,2,3,4,5 ONLY
------------------------------------------------
1 digit number

only 1 = 5

TOTAL = 1 way
-----------------------------------------------
2 digit numbers

possible sets (2,3) = 2*1 = 2 ways
if (4,1) then 2*1 = 2 ways
if (5,0) then = 1 way

TOTAL = 5 ways
------------------------------------------
3 digit numbers

possible sets (5,0,0) then =1 way

if (4,0,1)
2*2*1 = 4 ways

if one is 3 then 2 sets possible
(3,0,2) = 2*2*1 = 4 ways

(3,1,1) = 3*2*1/2 = 3 ways

if one is 2 then set can be (2,2,1) = 3*2*1/2 = 3 ways ... NOTE- we have already used a set of (2,3,0)

TOTAL = 1+4+4+3+3 = 15 ways

---------------------------------------------------

4 digits

possible sets (1,2,2,0) (1,1,1,2) (1,3,1,0) (3,2,0,0) (4,1,0,0) (5000)

_ _ _ _

3*3*2*1 /2 = 9 ways

4*3*2*1/6 = 4 ways

3*3*2*1/2 = 9 ways

2*3*2*1/2 = 6 ways

2*3*2*!/2 = 6 ways

1 way

respectively

TOTAL = 18+16+1 = 35 ways
--------------------------------------------------------
GRAND TOTAL = 35+21 = 56

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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 Sep 18, 2011 6:38 am

saketk wrote:
Brent@GMATPrepNow wrote:This one is well past the 800-level, but there are a lot of clever people on this site who will, I imagine, accept the challenge.

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
Answer: C

Try to resist examining each case and counting all of the possibilities. To discourage this strategy I could have asked "How many positive integers less than 1,000,000 are there in which the sum of the digits equals 8?"
Hi Brent -- Got this link from your latest post..

This is how I did it (though I used bit of a counting here)

------------------------------------------------
We can make numbers from ---0,1,2,3,4,5 ONLY
------------------------------------------------
1 digit number

only 1 = 5

TOTAL = 1 way
-----------------------------------------------
2 digit numbers

possible sets (2,3) = 2*1 = 2 ways
if (4,1) then 2*1 = 2 ways
if (5,0) then = 1 way

TOTAL = 5 ways
------------------------------------------
3 digit numbers

possible sets (5,0,0) then =1 way

if (4,0,1)
2*2*1 = 4 ways

if one is 3 then 2 sets possible
(3,0,2) = 2*2*1 = 4 ways

(3,1,1) = 3*2*1/2 = 3 ways

if one is 2 then set can be (2,2,1) = 3*2*1/2 = 3 ways ... NOTE- we have already used a set of (2,3,0)

TOTAL = 1+4+4+3+3 = 15 ways

---------------------------------------------------

4 digits

possible sets (1,2,2,0) (1,1,1,2) (1,3,1,0) (3,2,0,0) (4,1,0,0) (5000)

_ _ _ _

3*3*2*1 /2 = 9 ways

4*3*2*1/6 = 4 ways

3*3*2*1/2 = 9 ways

2*3*2*1/2 = 6 ways

2*3*2*!/2 = 6 ways

1 way

respectively

TOTAL = 18+16+1 = 35 ways
--------------------------------------------------------
GRAND TOTAL = 35+21 = 56

That looks great, saketk!
Lot of solid counting techniques here.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

tuanquang269: Master | Next Rank: 500 Posts; Posts: 296; Joined: Sun May 29, 2011 5:10 amFollowed by: 5 members; Location: Vietnam; Thanked: 10 times

by tuanquang269 » Wed Sep 21, 2011 6:38 pm

saketk wrote: 4 digits

possible sets (1,2,2,0) (1,1,1,2) (1,3,1,0) (3,2,0,0) (4,1,0,0) (5000)

_ _ _ _

3*3*2*1 /2 = 9 ways

4*3*2*1/6 = 4 ways

3*3*2*1/2 = 9 ways

2*3*2*1/2 = 6 ways

2*3*2*!/2 = 6 ways

1 way

Can you explain for me the way you count each set? I do not understand. For example (1,2,2,0) = 3 * 3 * 2 *1/2 (I don't understand where "3", "3", "2", "1/2" from?)

sana.noor: Legendary Member; Posts: 512; Joined: Mon Jun 18, 2012 11:31 pmFollowed by: 20 members; Thanked: 42 times

by sana.noor » Sun Jul 21, 2013 10:49 am

using the inserting stick and seperator technique

(5+3)!/5!.3! = 56...C is the answer

is my approach to answer this question right?

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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 Jul 21, 2013 10:51 am

sana.noor wrote:using the inserting stick and seperator technique

(5+3)!/5!.3! = 56...C is the answer

is my approach to answer this question right?

Perfect!!!

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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