Exercise 8. The goal of this exercise is to find the number of integer solutions of the equation x1 + x2 + x3 + x4 = 12 such that xi ? 0, i = 1, 2, 3 and 4. Suppose that you have 12 marbles to place in 4 boxes (x1 is the number of marbles put in box 1, x2 is the number of marbles put in box 2,...). One way to do this is, as shown in the picture, x1 = 1,x2 = 3, x3 = 4 and x4 = 4

Now, if we denote the marbles of the picture by zeros delimited by ones in a bit string of length 15 , we get: 0 1 000 1 0000 1 0000

Use your solution of Exercise 7 to solve the equation x1 + x2 + x3 + x4 = 12 where xi ? N ? {0}

Exercise 9. How many integer solutions of the equation x1 + x2 + x3 = 12 such that xi ? 1, ?i ? {1, 2, 3}.