Bhisham C. Gupta - Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP

Suppose that we have n distinct objects . We can determine how many different sequences of x objects can be formed by choosing x objects in succession from the n objects where . For convenience, we may think of a sequence of x places that are to be filled with x objects. We have n choices of objects to fill the first place. After the first place is filled, then with objects left, we have choices to fill the second place. Each of the n choices for filling the first place can be combined with each of the choices for filling the second place, thus yielding ways of filling the first two places. By continuing this argument, we will see that there are ways of filling the x places by choosing x objects from the set of n objects. Each of these sequences or arrangements of x objects is called a permutation of x objects from n . The total number of permutations of x objects from n , denoted by , is given by

(3.4.1)

Note that the number of ways of permuting all the objects is given by

(3.4.2)

where is read as n factorial.

Expressed in terms of factorials, we easily find that

(3.4.3)

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.