Question

How many solutions does the equation have, x1 + x2 + x3 = 10 , where x1 , x2, and x3 are non-negative integers?

Answer 1

Non-negative integers are positive integers or zero.

1. When $x_1=0,$ then there are such possible cases for $x_2$ and $x_3:$

• $x_2=0,\ x_3=10;$
• $x_2=1,\ x_3=9;$
• $x_2=2,\ x_3=8;$
• $x_2=3,\ x_3=7;$
• $x_2=4,\ x_3=6;$
• $x_2=5,\ x_3=5;$
• $x_2=6,\ x_3=4;$
• $x_2=7,\ x_3=3;$
• $x_2=8,\ x_3=2;$
• $x_2=9,\ x_3=1;$
• $x_2=10,\ x_3=0.$

In total 11 solutions for $x_1=0.$

2. For $x_1=1,$ there are such possible cases for $x_2$ and $x_3:$

• $x_2=0,\ x_3=9;$
• $x_2=1,\ x_3=8;$
• $x_2=2,\ x_3=7;$
• $x_2=3,\ x_3=6;$
• $x_2=4,\ x_3=5;$
• $x_2=5,\ x_3=4;$
• $x_2=6,\ x_3=3;$
• $x_2=7,\ x_3=2;$
• $x_2=8,\ x_3=1;$
• $x_2=9,\ x_3=0.$

In total 10 solutions for $x_1=1.$

3. This process gives you

• for $x_1=2$ - 9 solutions;
• for $x_1=3$ - 8 solutions;
• for $x_1=4$ - 7 solutions;
• for $x_1=5$ - 6 solutions;
• for $x_1=6$ - 5 solutions;
• for $x_1=7$ - 4 solutions;
• for $x_1=8$ - 3 solutions;
• for $x_1=9$ - 2 solutions;
• for $x_1=10$ - 1 solution.

4. Add all numbers of solutions:

$11+10+9+8+7+6+5+4+3+2+1=66.$

Answer: there are 66 possible solutions (with non-negative integer variables)

Answer 2

Answer:

There are total 66 solutions of the equations.

Step-by-step explanation:

We can find the solution by fixing the first value i.e. $x_1$ at a time and shifting the other two values and keep on doing for all the possible values of $x_1$

Hence, the possible cases are as follows:

$x_1-x_2-x_3$

0-10-0      1-9-0      2-8-0      3-0-7     4-0-6    5-0-5      6-0-4    

0-0-10      1-0-9      2-0-8      3-7-0     4-6-0     5-5-0     6-4-0

0-1-9         1-1-8       2-1-7       3-1-6      4-1-5      5-1-4       6-1-3

0-9-1         1-8-1       2-7-1       3-6-1      4-5-1      5-4-1       6-3-1

0-2-8        1-2-7      2-3-5      3-2-5      4-2-4     5-2-3      6-2-2

0-8-2        1-7-2      2-4-3      3-5-2      4-4-2     5-3-2

0-3-7         1-3-6      2-3-4      3-3-4      4-3-3

0-7-3         1-6-3      2-2-6      3-4-3

0-4-6         1-4-5      2-6-2

0-6-4         1-5-4

0-5-5

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7-0-3        8-0-2       9-0-1     10-0-0

7-3-0        8-2-0       9-1-0

7-1-2          8-1-1

7-2-1