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 Post subject: Combinatorics questions.Posted: Fri, 20 Feb 2009 09:50:36 UTC
 S.O.S. Newbie

Joined: Sun, 8 Feb 2009 00:28:41 UTC
Posts: 2
I have three questions, each of increasing difficulty level.

I. Newbie
I have a string of 7 digits. Zeros are allowed. The digits must always sum to 8. How many combinations of strings are there?

Examples
(1111112), 1+1+1+1+1+1+2=8
(8000000), 8+0+0+0+0+0+0=8
(2222000), 2+2+2+2+0+0+0=8
(1020311), 1+0+2+0+3+1+1=8

II. Intermediate.
I have string of n digits. Zeros are allowed. The digits must always sum to 8. How many combinations of strings are there?

III. Nightmare level.
I have string of n digits. Zeros are allowed. The digits must always sum to K. How many combinations of strings are there?

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 Post subject: Re: Combinatorics questions.Posted: Fri, 20 Feb 2009 10:31:03 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7573
Location: NCTS/TPE, Taiwan
Spoiler:
Assuming your base is >K. Biject with strings consisting of K 1's and n-1 separators. So (K+n-1 choose K).

Much harder is where your base is ≤K.

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 Post subject: Re: Combinatorics questions.Posted: Fri, 20 Feb 2009 16:10:03 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sun, 24 Jul 2005 20:12:39 UTC
Posts: 4875
Location: Ottawa Ontario
voskresiniye wrote:
I have a string of 7 digits. Zeros are allowed. The digits must always sum to 8. How many combinations of strings are there?

3003

from 1 to 9:
1,9,45,165,495,1287,3003,6435,12870

Go here and enter above:
http://www.research.att.com/~njas/sequences/index.html

_________________
I'm just an imagination of your figment...

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 Post subject: Re: Combinatorics questions.Posted: Fri, 27 Feb 2009 21:09:36 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 8286
Location: Lexington, MA
[delete]

Last edited by Soroban on Fri, 27 Feb 2009 21:30:02 UTC, edited 1 time in total.

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 Post subject: Re: Combinatorics questions.Posted: Fri, 27 Feb 2009 21:29:30 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 8286
Location: Lexington, MA
Hello, voskresiniye!

I have an approach to this problem . . . but that's all.

Quote:
I. I have a string of 7 digits. Zeros are allowed. The digits must always sum to 8.
How many combinations of strings are there?

Consider a row of eight blocks . . . with spaces before, between and after them.

. .

We have six "dividers" to place in any of the nine spaces.
Each configuration determines a string.

Examples

. .

. .

. .

. .

I would say that there are: . strings,
. . but I know that these are not unique strings.

I am presently trying to eliminate the duplication . . .

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