Fielding practice The coach of a Little League team made up

Fielding practice

The coach of a Little League team made up a game called fielding practice. A player got 10 points for catching a pop fly and making a good throw, 8 points for catching a pop fly and making a bad throw, 7 points for fielding a grounder and making a good throw, 5 points for fielding a grounder and making a bad throw, and 1 point for a good throw after making a catching error ( on either a pop fly or grounder). He scored 20 points in this game. In how many ways could he have scored 20 points?

Solution

10 points for catching a pop fly and making a good throw
8 points for catching a pop fly and making a bad throw
7 points for fielding a grounder and making a good throw
5 points for fielding a grounder and making a bad throw
1 point for a good throw after making a catching error

Total = 20

10a + 8b + 7c + 5d + e = 20

We need to find the total number of non-negative integral solutions to this one....

WE can have 2,0,0,0,0
WE can have 1,1,0,0,2
1,0,1,0,3
1,0,0,1,5
1,0,0,2,0
1,0,0,0,10
0,2,0,0,4
0,1,1,1,0
0,1,1,0,5
0,1,0,1,7
0,1,0,2,2
0,1,0,0,12
0,0,2,1,1
0,0,2,0,6
0,0,1,2,3
0,0,1,1,8
0,0,1,0,13
0,0,0,4,0
0,0,0,3,5
0,0,0,2,10
0,0,0,1,15
0,0,0,0,20

so, we have 22 ways!