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when p ≥ -3, q ≥ 0, r ≥ -2, and s ≥ -1

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when p ≥ -3, q ≥ 0, r ≥ -2, and s ≥ -1

by sanju09 » Sat Sep 25, 2010 4:40 am
How many integer solutions are there to p + q + r + s = 15, when p ≥ -3, q ≥ 0, r ≥ -2, and s ≥ -1?
(A) 24! / (21! 3!)
(B) 21! / (18! 3!)
(C) 18! / (15! 3!)
(D) 15! / (12! 3!)
(E) 12! / (9! 3!)



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by Night reader » Mon Feb 14, 2011 8:45 pm
this one has been waiting for five months to be answered :)
I will try my best; p ≥ -3, q ≥ 0, r ≥ -2, and s ≥ -1 ----> p+q+r+s ≥ -6
the possible arrangements in the interval {-6;15} are 21C3 (excluding q≥0)
IOM [spoiler]B 21! / (18! 3!)[/spoiler]
sanju09 wrote:How many integer solutions are there to p + q + r + s = 15, when p ≥ -3, q ≥ 0, r ≥ -2, and s ≥ -1?
(A) 24! / (21! 3!)
(B) 21! / (18! 3!)
(C) 18! / (15! 3!)
(D) 15! / (12! 3!)
(E) 12! / (9! 3!)



[spoiler]Took inspiration from: https://www.csee.umbc.edu[/spoiler]
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by fskilnik@GMATH » Tue Feb 15, 2011 1:56 am
sanju09 wrote:How many integer solutions are there to p + q + r + s = 15, when p ≥ -3, q ≥ 0, r ≥ -2, and s ≥ -1?
(A) 24! / (21! 3!)
(B) 21! / (18! 3!)
(C) 18! / (15! 3!)
(D) 15! / (12! 3!)
(E) 12! / (9! 3!)
Hi there!

Beautiful problem, sanju.

It is well know ("balls and separators argument") that the number of positive integer solutions to the equation p´+ q´ + r´ + s´ = 25 is given by C(25-1, 4-1) and that coincides with alternative (A). We will show below that this is indeed the solution to the original problem posted.

Please note that:

1) p >= -3 if and only if p´= p+4 >=1
2) q >= 0 if and only if q´= q+1 >=1
3) r >= -2 if and only if r´= r+3 >=1
4) s >= -1 if and only if s´= s+2 >=1

From the fact that p+q+r+s = 15 if and only if p´+ q´ + r´ + s´ = (p+4) + (q+1) + (r+3) + (s+2) = 15 + 10 = 25, we are done.

Regards,
Fabio.
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