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 Post subject: SUMmer
PostPosted: Mon, 5 Sep 2011 19:16:46 UTC 
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All numbers used are positive integers, including 0.
w numbers add up to z.
Of these, the first u numbers add up to x (w > u).

Example: w=7, z=15, u=3, x=6
We can say a+b+c+d+e+f+g = 15 and a+b+c = 6.

Question would be:
a+b+c+d+e+f+g = 15
a+b+c = 6
All are integers => 0
How many solutions?

One solution would be:
1 + 0 + 5 + 3 + 0 + 1 + 5 = 15
Another:
6 + 0 + 0 + 0 + 0 + 0 + 9 = 15

OK; if v = w-u and y = z - x
what is the general case solution in terms of u,v,x,y?

It has been my pleasure to devise that for your enjoyment :roll:

HINT: solution to the example is 6160.

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 Post subject: Re: SUMmer
PostPosted: Tue, 6 Sep 2011 00:11:11 UTC 
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So if 100 numbers add up to 1000 and the first 37 add up to 719 the answer is 67,152,065,523,364,994,999,519,574,379,210,511,437,523,335,458,111,588,334,827,065,725,071,506,439,116,752,777,366,145,675,169,765,503,864,386,964,859,987,888,310,842,234,600.
Is that right?


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 Post subject: Re: SUMmer
PostPosted: Tue, 6 Sep 2011 02:24:33 UTC 
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aswoods wrote:
So if 100 numbers add up to 1000 and the first 37 add up to 719 the answer is 67,152,065,523,364,994,999,519,574,379,210,511,437,523,335,458,111,588,334,827,065,725,071,
506,439,116,752,777,366,145,675,169,765,503,864,386,964,859,987,888,310,842,234,600.
Is that right?

Hmmm....I get answer of same length (131) but a bit lower:
63,954,348, ...... , 611,652,000 : me
67,152,065, ...... , 842,234,600 :you

You're making me nervous!!

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 Post subject: Re: SUMmer
PostPosted: Tue, 6 Sep 2011 03:11:58 UTC 
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Nope you're right... sorry bout that :lol:


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 Post subject: Re: SUMmer
PostPosted: Tue, 6 Sep 2011 04:08:30 UTC 
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Phewwwwww......
SO:
if v = w-u and y = z - x
what is the general case solution in terms of u,v,x,y?

You need to answer that to get a "pass mark" :wink:

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 Post subject: Re: SUMmer
PostPosted: Tue, 6 Sep 2011 04:22:52 UTC 
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Would it be...
Spoiler:
{x+u-1 \choose u-1}{y+v-1 \choose v-1}
?


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 Post subject: Re: SUMmer
PostPosted: Tue, 6 Sep 2011 06:06:39 UTC 
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C'est parfait, Monsieur Woods :idea:

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