Help Me To Find The Number Of Equation Solutions

Help me to solve this problem:

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

#Combinatorics #MathProblem #Math

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

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

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

gopinath no - 7 years, 8 months ago

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

Thanks!

Pranav Arora - 7 years, 8 months ago

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

Tim Vermeulen - 7 years, 8 months ago

@Tim Vermeulen – the book is good thanks Tim

Mashrur Fazla - 7 years, 8 months ago

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

Logan Dymond - 7 years, 8 months ago

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 - 7 years, 8 months ago

@Gopinath No – Thank u

Mahdi Al-kawaz - 7 years, 8 months ago

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

Ramakrishna Salagrama - 7 years, 8 months ago

@Ramakrishna Salagrama – Thanks for this link

Mashrur Fazla - 7 years, 8 months ago

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 - 7 years, 8 months ago

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

Firdous Fatima - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$