1-The Basic Principle of Counting, Permutations, Combinations.

The basic principle of counting will be fundamental to all our work.

It states that if one experiment can result in any of m possible outcomes and if another experiment can

result in any of n possible outcomes, then there are m n possible outcomes of the two experiments 

The basic principle of counting

Suppose  that  two  experiments  are  to  be  performed. 

Then  if experiment 1 can result in any one of m possible outcomes and if, for each  outcome  of  experiment  1,  there  are n

possible  outcomes  of experiment 2, then together there are m n possible outcomes of the two experiments.

A  small  community  consists  of  10  women,  each  of  whom  has  3 children. If one woman and one of her children are to be

chosen as mother and child of the year, how many different choices are possible?

By regarding the choice of the woman as the outcome of the first experiment and the subsequent choice of one of her children as

the outcome of the second experiment, we see from the basic principle that there are 10 * 3 = 30 possible choices.

When there are more than two experiments to be performed, the basic principle can be generalized.

The generalized basic principle of counting

If r experiments that are to be performed are such that the first one may result in any of 

possible outcomes; and if, for  each  of  these possible  outcomes,  there  are   possible  outcomes  of 

the  second experiment; and if, for each of the possible outcomes of the first two experiments, there are  

possible outcomes of the third experiment;

and if . . . , then there is a total of possible outcomes of the r experiments. 

How many  different  batting orders  are possible for  a  baseball team consisting of 9 players? 

There are 9! = 362,880 possible batting orders.

A  class  in  probability  theory  consists  of  6  men  and  4  women.  An examination is given, and the students

are ranked according to their performance. Assume that no two students obtain the same score.

(a) How many different rankings are possible?

(b) If the men are ranked just among themselves and then

the women just among themselves, how many different rankings are possible? 

(a) Because each ranking corresponds to a particular ordered arrangement of the 10 people, the answer

to this part is  10! = 3,628,800.

(b) Since there are 6! possible rankings of the men among themselves and 4!

possible rankings of the women among themselves, it follows

from the basic principle that there are (6!)(4!) = (720)(24) = 17,280 possible rankings in this case.

Ms. Jones has 10 books that she is going to put on her bookshelf.

Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book.

Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the shelf.

How many different arrangements are possible? 

There are 4! 3! 2! 1! arrangements such that the mathematics books are first in line, then the chemistry books, then

the history books, and then the language book.

Similarly, for each possible ordering of the subjects, there are 4! 3! 2! 1! Possible arrangements.

Hence, as there are 4! possible orderings of the subjects, the desired answer is  4! 4! 3! 2! 1! = 6912.

We shall now determine the number of permutations of a set of n objects when

certain of the objects are indistinguishable from each other.

To set this situation straight in our minds, consider the following example.

How many different letter arrangements can be formed from the letters PEPPER? Solution.

We  first note  that there are 6!  permutations  of  the letters when

the 3P’s and the 2E’s are distinguished from each other. However, considerany one of these

permutations—for instance, . If we now permute the P’s among themselves and the E’s among

themselves, then the resultant arrangementwould still be of the form PPEPER. That is, all 3! 2! Permutations

are of the form PPEPER. Hence, there are 6!/(3! 2!) = 60 possible

letter arrangements of the letters PEPPER.  In general, the same reasoning as that

used in Example 5 shows that there are

different permutations of n objects, of which are alike,  are alike,are alike. 

A chess tournament has 10 competitors, of which 4 are Russian, 3 are from

the United States, 2 are from Great Britain, and 1 is from Brazil.

If the tournament result lists just the nationalities of the players in

the order in which they placed, how many outcomes are possible? Solution.

There are =12,600 possible outcomes. 

How many different signals, each consisting of 9 flags hung in a line, can be

made from a set of 4 white flags, 3 red flags, and 2 blue flags if all

flags of the same color are identical? Solution. There are =1260 different signals.