6. When being tested, an integrated circuit (IC) is considered as a black box that performs certain designed functions. Four ICs will be randomly selected from a shipment of 15 and will be tested for static voltages, external components associated with the IC, and dynamic operation. Let X1, X2, and X3 be the number of ICs in the sample that fail the first, second, and third test, respectively, and X4 be the number of ICs in the sample that do not fail any of the tests. Suppose that, if tested, three of the 15 ICs would fail only the first test, two would fail only the second test, one would fail only the third test, and nine would not fail any of the three tests.

(a) Specify the sample space of (X1, . . . ,X4).

(b) Find the joint PMF of X1, X2, and X3.