Help me to solve this problem:

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

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

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## Comments

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TopNewestIf one number is thrice the other and their sum is 16 find the numbers

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

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

\([z^{14}]G(z)=\binom{17}{3}+\binom{12}{3}+\binom{7}{3} = 935\)

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This looks good. Can you please share a link which explains generating functions and their applications? I have often seen people using this.

Thanks!

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Generatingfunctionology is a great book for that, you can (legally) download the second edition of that book here.

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

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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.

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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

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