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March 22nd, 2011, 03:03 PM   #1
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permutation of a sum of ranges

Hi,
How to find the permutations/combinations of the possible values of items such as the sum of these is constant,
but each item's value may vary in the range 0-20? F.ex:

[color=#800040]A1(0-20)+A2(0-20)+A3(0-20)+A4(0-20)=constant(20)[/color]

I could only get the problem down to:

[color=#800000]A1(0-20)+A2(0-20)=20 [/color]

and the combinations/permutations (not even sure what it is about exactly, cause when A1 goes to 20, A2 goes down to 0)
are obviously 20, derived just from intuitive logic.

Cannot think off a way to utilize the formulas for combinations/permutations
[color=#400040]P(n, r) = n!/(n - r)!, C(n, r) = n!/r!(n - r)![/color]

Best Regards
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March 22nd, 2011, 11:08 PM   #2
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Re: permutation of a sum of ranges

You mean 21, because the first number will be 0-20 and the second number 20-0, and there are 21 possibilities.
In general, if you need two non-negative integers to sum to N, there will be N+1 ways.

So if you have four numbers, you can look at them as two pairs.
If the first pair sums to 20 (in 21 ways), the second pair sums to 0 (in 1 way).
If the first pair sums to 19 (in 20 ways), the second pair sums to 1 (in 2 ways).
If the first pair sums to 18 (in 19 ways), the second pair sums to 2 (in 3 ways).
...

21*1 + 20*2 + 19*3 + 18*4 + ... + 11*11 + ... + 4*18 + 3*19 + 2*20 + 1*21
= (21 + 40 + 57 + 72 + 85 + 96 + 105 + 112 + 117 + 120) * 2 + 121
= 825*2 + 121
= 1771

I could give you a general formula (for four numbers) if you need one.
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March 23rd, 2011, 12:55 AM   #3
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Re: permutation of a sum of ranges

Formula with something! factorial?
Still cant get it. Represented as sets isn't it rather:
{20,0},{19,1},{18,2},{17,3},{16,4},{15,5},{14,6},{ 13,7},{12,8},{11,9},
{10,10},{9,11},{8,12},{7,13},{6,14},{5,15},{4,16}, {3,17},{2,18},{1,19},
{0,20} = 21 sets, ok I get that a + b = N in N + 1 ways but still..
Why do you multiply them?
Also each item should occupy once the range 0-20: A1(0-20), A2(0-20)...
So when you combine the results of these 2x2 pairs, what exactly is going on?
maybe a formula for n items that sum up to N, as:
I(n-(n-1)) + I(n-(n-2)) + ... + I(n) = N
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March 23rd, 2011, 01:07 AM   #4
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Re: permutation of a sum of ranges

Suppose the first two sum to 18. There are 19 ways it could happen:

18+0
17+1
16+2
etc

Then for each of the ways it could happen, there are 3 ways the other two could sum to 2.

So you would have
18+0 + 2+0
18+0 + 1+1
18+0 + 0+2

17+1 + 2+0
17+1 + 1+1
17+1 + 0+2

16+2 + 2+0
16+2 + 1+1
16+2 + 0+2

etc
19*3 ways
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March 23rd, 2011, 01:29 AM   #5
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Re: permutation of a sum of ranges

Got it more or less. Do we need an induction proof here?
Is it mathematically correct to leave it like this for this particular case (4 items sum up to 20)
or:
It(1) + It(2) + ... + It(n) = N where It(0 ... N)

a formula for n Items that sum up to N proved by induction
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