Help me to solve this problem:

Find the number of non-negative integer solution of the equation:

\(5x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=14\)

Help me to solve this problem:

Find the number of non-negative integer solution of the equation:

\(5x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=14\)

No vote yet

5 votes

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestIt's the coefficient of z^14 in the generating function

\(G(z)=\dfrac{1}{(1-z^5)\,(1-z)^4}\)

\([z^{14}]G(z)=\binom{17}{3}+\binom{12}{3}+\binom{7}{3} = 935\) – Gopinath No · 3 years, 10 months ago

Log in to reply

Thanks! – Pranav Arora · 3 years, 10 months ago

Log in to reply

here. – Tim Vermeulen · 3 years, 10 months ago

Generatingfunctionology is a great book for that, you can (legally) download the second edition of that bookLog in to reply

– Logan Dymond · 3 years, 10 months ago

Tim, what a great guy, I'm glad that you respect copyrights. Thank you!Log in to reply

– Mashrur Fazla · 3 years, 10 months ago

the book is good thanks TimLog in to reply

This file covers some of the interesting things that can be done using GF, and go through the references, which are good enough, I think. – Gopinath No · 3 years, 10 months ago

Log in to reply

– Mahdi Al-kawaz · 3 years, 10 months ago

Thank uLog in to reply

– Ramakrishna Salagrama · 3 years, 10 months ago

You can learn some concepts form this article http://www.campusgate.co.in/2013/09/integer-solutions-using-coefficient.htmlLog in to reply

– Mashrur Fazla · 3 years, 10 months ago

Thanks for this linkLog in to reply

Case I x1= 0

x2 + x3 + x4 + x5 =14

Above is a linear Diophantine equation.

The number of non-negative solution to above equation is given by

(14+4-1)C(4-1) = 17C3 = 680

Case II x1= 1

x2 + x3 + x4 + x5 =9

The number of non-negative solution to above equation is = 12C3 = 220

Case III x1= 2

x2 + x3 + x4 + x5 =4

The number of non-negative solution to above equation is =7C3 = 35

Hence total number of non-negative integer solutions = 680 + 220 +30 = 935 – Ram Prakash Patel Patel · 3 years, 10 months ago

Log in to reply

If one number is thrice the other and their sum is 16 find the numbers – Firdous Fatima · 3 years, 9 months ago

Log in to reply