2019 AMC 10C Problems/Problem 11

Solution

We could use casework:

1 1 1 1 1 1 1 1 1 1 = 10 choose 0 = 1 case

1 1 1 1 1 1 1 1 2 = 9 choose 1 = 9 cases

1 1 1 1 1 1 2 2 = 8 choose 2 = 28 cases

1 1 1 1 2 2 2 = 7 choose 3 = 35 cases

1 1 2 2 2 2 = 6 choose 2 = 15 cases

2 2 2 2 2 = 5 choose 0 = 1 case

We add up 1 + 9 + 28 + 35 + 15 + 1 = cases.