Can someone help with these kind of questions
Find the number of integral solution of the following equation:
$$|x|+|y|+|z|=5$$
Asked
Active
Viewed 132 times
-1
-
where can i plot the integer solutions? – Dr. Sonnhard Graubner Jan 09 '17 at 16:30
-
see this: https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29 – Arnaldo Jan 09 '17 at 16:35
2 Answers
3
Hints:
$x$ and $y$ can take any the integer values $0,\pm1,\pm2,\pm3,\pm4,\pm5$; they can't take any other values (why not?). The value of $|z|$ is then determined by those of $x$ and $y$; as $z$ also must be one of the integer values listed, you must have $|x|+|y|\le5$. If $|x|+|y|=5$, then $z=0$, otherwise $z$ has two values, one positive and one negative.
Using these hints, can you make a start?
George Law
- 4,103
0
This is an application of the stars-and-bars theorem.
Check this out : How to use stars and bars (combinatorics)
In this case, your problem is asking for the number of non-negative integral solutions to the equation.
$$ x+y+z=5 $$ The answer will be $\binom{5+3-1}{5}=21$
Vishnu V.S
- 975