How many ways are there to pick 10 balls from a pile of ball

How many ways are there to pick 10 balls from a pile of balls that has lots of red, blue and white balls and also one lavender, one pink, and one tan ball?

Solution

if all three of lavender pink and tan are chosen then number of cases

3C3 (choosing all 3)* 9C2(no of solution to x_1 +x_2+x_3 = 7)=36

if 2 out of three of lavender pink and tan are chosen then number of cases

3C2 (choosing all 3)* 10C2(no of solution to x_1 +x_2+x_3 = 8) = 135

if 1 out of three of lavender pink and tan are chosen then number of cases

3C1 (choosing all 3)* 11C2(no of solution to x_1 +x_2+x_3 = 9) = 165

if none out of three of lavender pink and tan is chosen then number of cases

3C0 (choosing all 3)* 12C2(no of solution to x_1 +x_2+x_3 = 9) = 66