ho many integer solutions are there to the equation abc21 if

ho many integer solutions are there to the equation a+b+c=21 if a \\geq 3 b \\geq 1 and c \\geq 2?

Solution

We can rewrite the equation as

(a-3)+(b-1)+ (c-2)=15

Or

x1+x2+x3=15 ,x1=a-3,x2=b-1,x3=c-2

x1>=0,x2>=0,x3>=0

So problem reduces to distributing 15 objects to 3 people each can take 0 or more

This is done in

C(15+3-1,3-1)=C(17,2)=136