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What is the number of non-negative integer solutions of the equation px+qy+rzgreater or equal thanm, if p, q, r are constants and the maximum possible value of x is a, max. possible value of y is b, and max. possible value of z is c?

Asked by Aditya Great 19th March 2015, 9:05 PM
Answered by Expert
Answer:
The number of non-negative integral solutions of the equation
px + qy + rz = m is equivalent to coefficient of xm in the expansion of
open parentheses 1 plus x to the power of p plus x to the power of 2 p end exponent plus x to the power of 3 p end exponent plus.... plus x to the power of a p end exponent close parentheses open parentheses 1 plus x to the power of q plus x to the power of 2 q end exponent plus x to the power of 3 q end exponent plus.... plus x to the power of b q end exponent close parentheses open parentheses 1 plus x to the power of r plus x to the power of 2 r end exponent plus.... plus x to the power of c r end exponent close parenthesesequals open parentheses fraction numerator 1 minus x to the power of p left parenthesis a plus 1 right parenthesis end exponent over denominator 1 minus x to the power of p end fraction close parentheses open parentheses fraction numerator 1 minus x to the power of q open parentheses b plus 1 close parentheses end exponent over denominator 1 minus x to the power of q end fraction close parentheses open parentheses fraction numerator 1 minus x to the power of r open parentheses c plus 1 close parentheses end exponent over denominator 1 minus x to the power of r end fraction close parentheses equals open square brackets open parentheses 1 minus x to the power of p close parentheses open parentheses 1 minus x to the power of q close parentheses open parentheses 1 minus x to the power of r close parentheses close square brackets to the power of negative 1 end exponent open square brackets open parentheses 1 minus x to the power of p left parenthesis a plus 1 right parenthesis end exponent close parentheses open parentheses 1 minus x to the power of q open parentheses b plus 1 close parentheses end exponent close parentheses open parentheses 1 minus x to the power of r open parentheses c plus 1 close parentheses end exponent close parentheses close square brackets T h e space c o e f f i c i e n t space o f space x to the power of m space i n space t h e space a b o v e space e x p r e s s i o n space i s space e q u i v a l e n t space t o space c o e f f i c i e n t space o f space x to the power of m space i n space t h e space e x p a n s i o n space o f open square brackets open parentheses 1 minus x to the power of p close parentheses open parentheses 1 minus x to the power of q close parentheses open parentheses 1 minus x to the power of r close parentheses close square brackets to the power of negative 1 end exponent S i m i l a r l y space w e space f i n d space t h e space c o e f f i c i e n t space o f space x to the power of n space i n space t h e space e x p a n s i o n space o f space open square brackets open parentheses 1 minus x to the power of p close parentheses open parentheses 1 minus x to the power of q close parentheses open parentheses 1 minus x to the power of r close parentheses close square brackets to the power of negative 1 end exponent comma space w h e r e space n greater than m. space T h e n comma space w e space a d d space a l l space s u c h space c o e f f i c i e n t s space o b t a i n e d space t o space g e t space t h e space n u m b e r space o f space n o n minus n e g a t i v e space i n t e g r a l space s o l u t i o n s space o f space p x plus q y plus r z greater or equal than m.
Answered by Expert 23rd March 2015, 11:19 AM
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