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Help me to solve this problem:

Find the number of non-negative integer solution of the equation: $5x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=14$

Note by Mashrur Fazla 5 years, 10 months ago

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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!

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

@Tim Vermeulen – the book is good thanks Tim

@Tim Vermeulen – Tim, what a great guy, I'm glad that you respect copyrights. Thank you!

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 – Thank u

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

@Ramakrishna Salagrama – Thanks for this link

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

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

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Remember to wrap math in $</span> ... <span>$ or $</span> ... <span>$ to ensure proper formatting.`2 \times 3`

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`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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

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

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

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

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

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