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

Non-negative integers are positive integers or zero.

1. When then there are such possible cases for and

In total 11 solutions for

2. For there are such possible cases for and

In total 10 solutions for

3. This process gives you

4. Add all numbers of solutions:

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

There are total 66 solutions of the equations.

Step-by-step explanation:

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

Hence, the possible cases are as follows:

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