There are 5 types of balls in a box. The quantity of each ball present is $x_1,x_2,x_3,x_4,x_5$ ($x_1$ balls of 1st type, $x_2$ balls of 2nd type etc.) Now if we choose any one of the balls of type 1, 2, 3 or 4 all the balls of that chosen type get destroyed in the box. However if we choose a ball of type 5, all the balls (every type) in the box get destroyed.
I need to find the expected value of number of balls chosen until the box gets empty (all balls destroyed).
Is there an easier way to find the answer somehow using the Linearity of Expectation? Or do I need to find probability corresponding to each possible case?