Solutions to equation!?

Question

How many solutions are there to the equation x1 + x2+ x3 + x4 = 13

where x1, x2, x3, x4 are non-negative integers?

Full working needed.

Answer 1

Guess x1>=x2>=x3>=x4. Work systematically: 13 =
13 + 0 + 0 + 0 =
12 + 1 + 0 + 0 =
11 + 2 + 0 + 0 =
11 + 1 + 1 + 0 =
10 + 3 + 0 + 0 =
10 + 2 + 1 + 0 =
10 + 1 + 1 + 1 =
9 + 4 + 0 + 0 =
Go on yourself and count the number of possibilities at the and.
Good luck

Th

Answer 2

You didn't say they all had to be different integers, nor that the integers had to be greater than zero.

Four solutions right off the bat are

13-0-0-0
0-13-0-0
0-0-13-0
0-0-0-13

Twelve more solutions are

12-1-0-0
12-0-1-0
12-0-0-1
1-12-0-0
0-12-1-0
0-12-0-1
1-0-12-0
0-1-12-0
0-0-12-1
1-0-0-12
0-1-0-12
0-0-1-12

Solutions involving 11 would take the form

11-2-0-0
11-0-2-0
11-0-0-2
11-1-1-0
11-1-0-1
11-0-1-1

The are 24 solutions involving 11.