Plastic parts produced by an injection-molding operation are checked for conformance to specifications. Each tool contains 12 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An inspector chooses 3 parts from among the 12 at random. Two cavities are affected by a temperature malfunction that results in parts that do not conform to specifications.

a) How many samples contain exactly 1 nonconforming part?
b) How many samples contain at least 1 nonconforming part?

- For first part, we want the inspector to pick exactly one non-conforming part among 3 selected. So let us say that he has already chosen that one non conforming cavity. Now he has to make 2 more selections out of total conforming cavities = 12 - 2 = 10 conforming cavities. Hence, the total possible outcome is to chose 2 randomly from 10 conforming cavities.

( Exactly 1 ) 10C2 = 45 possible outcomes

- The second part entails that at-least 1 non-conforming cavity is selected. To choose exactly 1 non conforming we calculated above. In the similar way calculate for selecting exactly 2 non-conforming cavities. The total possible outcome would be to choose from 10 conforming and we choose 1 from it:

( Exactly 2 ) 10C1 = 10 possible outcomes

- Hence, for at-least 1 non conforming cavity being selected we same the above two cases calculated:

(At-least 1 ) = ( Exactly 1 ) + ( Exactly 2 )

(At-least 1 ) = 45 + 10 = 55 possible outcomes