Above 700 question!!!

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Ozlemg: Master | Next Rank: 500 Posts; Posts: 407; Joined: Tue Jan 25, 2011 9:19 amFollowed by: 7 members; Thanked: 25 times

by Ozlemg » Fri Jul 01, 2011 5:04 am

How many positive integers less than 10,000 are there in which sum of digits equals 5?
(a) 31
(b) 51
(c) 56
(d) 62
(e) 93

The more you suffer before the test, the less you will do so in the test!

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Source: Beat The GMAT — Problem Solving | Fri Jul 01, 2011 5:04 am

Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Fri Jul 01, 2011 5:32 am

Hi,
Let abcd be the number. We need to find the number of non-negative integral solutions of a+b+c+d = 5
It is (5+4-1)C4-1 = 8C3 = 56
Hence, C

If x1+x2+..+xk = n, then the number of non-negative solutions is (n+k-1)C(k-1).

Cheers!

Things are not what they appear to be... nor are they otherwise

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Ozlemg: Master | Next Rank: 500 Posts; Posts: 407; Joined: Tue Jan 25, 2011 9:19 amFollowed by: 7 members; Thanked: 25 times

by Ozlemg » Fri Jul 01, 2011 5:38 am

How did you get that it is a combination problem?
I could not! Does this kind of problems need "combination" approach?

The more you suffer before the test, the less you will do so in the test!

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Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Fri Jul 01, 2011 5:58 am

Ozlemg wrote:How did you get that it is a combination problem?
I could not! Does this kind of problems need "combination" approach?

This is not a combination problem directly. But, by solving we end up with this formula. This will save time, else you need to consider cases:
1)count 1 digit numbers - only one(5)
2)count 2 digit numbers - five(14,23,32,41,50)
3)count 3 digit numbers ...
4)count 3 digit numbers ...
5)count 3 digit numbers ...

Cheers!

Things are not what they appear to be... nor are they otherwise

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Fri Jul 01, 2011 7:19 am

He is a human for sure!

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Jul 01, 2011 9:08 am

Ozlemg wrote:How many positive integers less than 10,000 are there in which sum of digits equals 5?
(a) 31
(b) 51
(c) 56
(d) 62
(e) 93

Another approach is to use the separator method. Check here:

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

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Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Fri Jul 01, 2011 9:37 am

GMATGuruNY wrote: Another approach is to use the separator method. Check here:
https://www.beatthegmat.com/experts-any- ... 82307.html

Exactly. This separator method gives the formula (n+k-1)C(k-1).

Cheers!

Things are not what they appear to be... nor are they otherwise

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baladon99: Senior | Next Rank: 100 Posts; Posts: 66; Joined: Wed Jul 07, 2010 11:06 am; Thanked: 1 times

by baladon99 » Fri Jul 01, 2011 12:19 pm

the sum should be 5.maximum it can be a 4 digit number.

In the following cases the sum is 5.

1004 --> In 12 ways it can be represented ( 4!/2!)
1103---> 12 ways
2003---> 12 ways
2102---> 12 ways
1112----> 4 ways (4!/3!)
5000---> 4 ways

In total 56 ways

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