Probability and Statistical Inference. Robert Bartoszynski

Probability and Statistical Inference

the elements in the union images

, equal to images

Example 3.3 License Plates

Most states now use the system where a license plate has six symbols. One type (call it A) of such licenses has a prefix of three letters followed by a three‐digit number (e.g., CQX 786). Other states use system (call it B) with a two‐letter prefix, followed by a four‐digit number (e.g., KN 7207). Still other states use system C, a digit and two letters, followed by a three digit number (e.g., 2CP 412). In addition states allow (for a special fee) “personalized” plates such as CATHY3 or MYCAR. Disregarding the personalized plates, which type of the license plate system can register most cars?

Solution

Let images and images stand for the alphabet and for the set of 10 digits: images Then, a license plate from system A can be regarded as an element of images , while a license plate in system B is an element of the set images . The numbers of elements in these Cartesian products are images and images . The ratio is images , so in the state using an A system 2.6 times more cars can be registered than in the state with a B system.

Regarding system C, the answer depends whether or not 0 is allowed in the prefix. If the plate such as 0HY 314 is not allowed (e.g., because the digit 0 can be confused with the letter O), then the number of possible license plates is only images , which is 10% less than the number of plates possible in states using system B. If 0 is allowed as the first character, then the numbers of plates of types images and images are the same.

In Examples 3.1 through 3.3 the set of ways of performing the second operations is the same regardless of which option was selected for the first operation. However, Theorem 3.2.1 remains true if the sets of ways of performing the second operation depend on the choice of the first operation. In particular, we can think of the first and second operation as two consecutive choices of an element from the same set, without returning the chosen elements. If the set, say images , has images elements, then the first operation (choice of an element) can be performed in images ways. If the chosen element is not returned, then the second choice can be performed in images ways only, and we have the following:

Corollary 3.2.4 The number of ordered pairs images with images that can be formed out of images distinct elements of a set of size images is images .

Instead of thinking in terms of operations, we can still use Cartesian products here. Thus, images has images elements by Theorem 3.2.3, of which images are of the form images . The number of pairs with elements distinct is images .

We can generalize these considerations as follows:

Definition 3.2.1 An ordered sequence of images elements selected without replacement from a set of images distinct elements images is called a permutation of images out of images elements.

Theorem 3.2.5 The number of permutations of images out of images , denoted images , equals

Proof: The argument here repeatedly uses the “operation” principle: the first choice can be made in images ways, the second in images ways, the images th in images ways.

If images , consecutive choices form an ordering of the entire set of size images . We obtain the following:

Corollary 3.2.6 The set of images elements can be ordered in

(3.2) equation

distinct ways.

The product (3.2)

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