EXAMPLE 46 Acceptance sampling uses combinations
Suppose we have a batch of 20 cell phones, of which, unknown to us, 3 are defective and 17 are nondefective; we will take a random sample of size 2 and evaluate both items once.
- Are the arrangements in acceptance sampling permutations or combinations?
- Find the number of ways that both sampled cell phones are defective.
- Both permutations and combinations require the following:
- items are chosen from distinct items. Here, we are selecting phones from a batch of .
- Repetition of the items is not allowed. Each item is evaluated only once. The difference between permutations and combinations is that, for permutations, order is important, whereas for combinations, order is not important. In acceptance sampling, the order of the items is not important. Thus, acceptance sampling uses combinations.
The number of ways of choosing two of the three defectives is
Selecting 2 defectives means that we are choosing 0 of the 17 nondefectives. The number of ways this can happen is
By the Multiplication Rule for Counting, the number of ways that both sampled cell phones are defective is