How many integer solutions are there to the equation x1 x2

How many integer solutions are there to the equation x_1 + x_2 + x_3 + x_4 = 35 with x_1 x_2 > 4 and x_3, x_4 greaterthanorequalto 8?

x1 + x2 + x3 + x4 = 35

Let x1 = a1 + 5, and a1 >= 0 x2 = a2 + 5, and a2 >= 0

Let x3 = a3 + 8 and a3 >= 0 x4 = a4 + 8, and a4 >= 0

So, problem reduces to :

a1 + a2 + a3 + a4 = 35 - 10 - 16 = 9 where a1,a2,a3,a4 >= 0

For such problems: a1 + a2 + a3 .... + am =n , we have known formula: Combinations( n+m-1, m-1 )

So, answer is : Combinations(9 + 4 - 1, 4 -1 ) = Combinations( 12, 3 ) = (12*11*10)/(3*2) = 220