Читать книгу Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP - Bhisham C. Gupta, Irwin Guttman - Страница 122

It is easy to see that if we select any set of x objects from n, there are ways this particular set of x objects can be permuted. In other words, there are permutations that contain any set of x objects taken from the n objects. Any set of x objects from n distinct objects is called a combination of x objects from n objects. The number of such combinations is usually denoted by . As each combination of each x objects can be permuted in ways, these combinations give rise to permutations. But this is the total number of permutations when using x objects from the n objects. Hence, , so that

(3.4.4)

Example 3.4.2 (Applying concept of combinations) The number of different possible hands of 13 cards in a pack of 52 ordinary playing cards is the number of combinations of 13 cards from 52 cards, and from (3.4.4) is

Example 3.4.3 (Applying concept of combinations) The number of samples of 10 objects that can be selected from a lot of 100 objects is

Example 3.4.4 (Determining number of combinations) Suppose that we have a collection of n letters in which x are A's and are B's. The number of distinguishable arrangements of these n letters (x A's and B's) written in n places is .

We can think of all n places filled with B's, and then select x of these places and replace the B's in them by A's. The number of such selections is . This is equivalent to the number of ways we can arrange x A's and B's in n places.

The number is usually called binomial coefficient, since it appears in the binomial expansion (for integer )

(3.4.5)

Example 3.4.5 The coefficient of in the expansion of is , since we can write as

(3.4.6)

The coefficient of is the number of ways to pick x of these factors and then choose a from each factor, while taking b from the remaining factors.