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 June 5th, 2012, 06:20 PM #1 Member   Joined: May 2012 Posts: 86 Thanks: 0 Counting Problem Consider the equation x+y+z+w=n, where n is a positive integer greater or equal to 4. A positive and integer solution is a set (x,y,z,w) of positive integers that satisfies the equation for a given n. For example, for n=10 one solution to x+y+z+w=10 would be (1,2,3,4), since 1+2+3+4=10. a) Determine the number of positive and integer solutions for n=10. b) Find a general formula that counts the number of whole and integer solutions for x+y+z+w=n. Note: Solutions (2, 2, 2, 4), (2, 2, 4, 2),(2, 4, 2, 2) and (4, 2, 2, 2) for n=10 are each considered different and separate solutions. June 5th, 2012, 07:14 PM #2 Member   Joined: May 2012 From: Chennai,India Posts: 67 Thanks: 0 Re: Counting Problem so. its finding the no of partitions of n as 4 integers.. it can be split like partitions of n as sum of two numbers.. and finding the partitions of those 2 numbers as sum of two numbers... June 5th, 2012, 10:47 PM #3 Member   Joined: May 2012 From: Chennai,India Posts: 67 Thanks: 0 Re: Counting Problem this can be proved by multinomial theorem... For a+b+c+d = n, the no. of solutions including 0 is need to eliminate those terms in the expansion of that have powers of all a,b,c,d June 6th, 2012, 03:14 AM   #4
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Re: Counting Problem

Quote:
 Originally Posted by karthikeyan.jp this can be proved by multinomial theorem... For a+b+c+d = n, the no. of solutions including 0 is need to eliminate those terms in the expansion of that have powers of all a,b,c,d
What's the reasoning behind that formula? June 6th, 2012, 05:10 AM #5 Member   Joined: May 2012 From: Chennai,India Posts: 67 Thanks: 0 Re: Counting Problem The number of co-efficients in the expansion of is (n+m-1 n). http://en.wikipedia.org/wiki/Multinomial_theorem So for , it is (n+3 n) ==> June 6th, 2012, 05:40 AM   #6
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Re: Counting Problem

Quote:
 Originally Posted by karthikeyan.jp this can be proved by multinomial theorem... For a+b+c+d = n, the no. of solutions including 0 is
Well, using this formula, I found through trial and error that the general formula for my problem happens to be:
((n-3)(n-2)(n-1))/6

I'm completely clueless as to the reasoning behind it, though, and politely ask for a baby explanation lol. June 7th, 2012, 03:32 PM   #7
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Re: Counting Problem

Hello, Jakarta!

Quote:
 Consider the equation , where is a positive integer A solution is a set of positive integers that satisfies the equation for a given a) Determine the number of solutions for

Place 10 objects in a row, inserting a space between them.
[color=beige]. . [/color]

Select 3 of the 9 spaces and insert "dividers".

Quote:
 b) Find a general formula that counts the number of solutions for Note: Solutions (2, 2, 2, 4), (2, 2, 4, 2), (2, 4, 2, 2) and (4, 2, 2, 2) for [color=beige]. . . . . [/color]are each considered different and separate solutions.

Following the solution in part (a), place objects in a row (with spaces).

Select 3 of the spaces and insert "dividers". June 7th, 2012, 09:05 PM   #8
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Re: Counting Problem

Quote:
Originally Posted by soroban
Hello, Jakarta!

Quote:
 Consider the equation , where is a positive integer A solution is a set of positive integers that satisfies the equation for a given a) Determine the number of solutions for

Place 10 objects in a row, inserting a space between them.
[color=beige]. . [/color]

Select 3 of the 9 spaces and insert "dividers".

[quote:3oncju6n]b) Find a general formula that counts the number of solutions for

Note: Solutions (2, 2, 2, 4), (2, 2, 4, 2), (2, 4, 2, 2) and (4, 2, 2, 2) for
[color=beige]. . . . . [/color]are each considered different and separate solutions.

Following the solution in part (a), place objects in a row (with spaces).

Select 3 of the spaces and insert "dividers".

[/quote:3oncju6n]
Wow, that is extremely smart and clear. Thanks! Tags counting, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post superconduct Algebra 2 January 7th, 2014 10:01 AM zelmac Algebra 0 February 14th, 2013 05:29 AM scream Applied Math 2 February 21st, 2012 11:50 AM kec11494 Applied Math 1 December 20th, 2010 08:44 PM julian21 Applied Math 0 April 27th, 2010 09:48 AM

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