1. ENM 207 Lecture 5

2. FACTORIAL NOTATION The product of positive integers from 1 to n is denoted by the special symbol n! and read “n factorial”. n!=1.2.3….(n-2).(n-1).n ex: 5!=1.2.3.4.5=120

3. Some special factorial values We make the following mathamatical manipulation: Product and divide the left side of above equation by (n-r)! and obtain n!/(n-r)!

4. PERMUTATIONS Any ordered sequence of k objects taken from a set of n distinct obfects is called a permutation of size k of the objects. The number of permutations of size k is obtained from the general product rule as follows: The first element can be chosen in n ways, the second element can be chosen in n-1 ways, and so on ;

5. PERMUTATIONS Finally for each way of choosing the first k-1 elements, the k th element can be chosen in n-(k-1) = n-k+1 ways, thus The number of permutations of size k in n distinct object is denoted by

6. COMBINATIONS Given a set of n distinct objects any unordered subset of size k of the objects is called a combination. The number of combinations of size k that can be formed from n distinct objects will be denoted by

7. COMBINATIONS The number of combinations of size k from a particular set is smaller than the number of permutations because, when order is disregarded, a number of permutations correspond to the same combination.

8. COMBINATIONS Ex: consider the set {A,B,C,D,E} consisting of 5 elements. We know that there are 5!/(5-3)!=60 permutations of size 3 and 5!/ 3!(5-3)!= 10 combinations of size 3 Ex: find the number of permutations of size 3 consisting of the elements of A,B,C. 3! = 3 x 2 x 1 = 6 (A,B,C) (A,C,B) (B,A,C) (B,C,A) (C,A,B) and (C,B,A)

9. Ex: repititions are not permited How many 3 digit numbers can be formed from the six digits 2, 3, 5, 6,7 and 9? i) 654=120 numbers i) How many of these are less than 400? 254=40 numbers The box on the left can be filled in only two ways, by 2 or 3, since each number must be less than 400; The middle box can be filled in 5 ways. The box on the right can be filled in 4 ways.

10. repititions are not permited i) How many are even? Firs start filling from right side to provide condition. The box on the right can be filled in only 2 ways by 2 or 6, since the numbers must be even. The box on the left can be filled 5 ways. The box on the middle can be filled 4 ways 5 42

11. a) Theorem: Let A contain n elements and let n 1, n 2,,,,,, n r be positive integers with n 1 + n 2 + n 3 +,,,,,+ n r = n Lets A 1, A 2,...., A r are different partitions of A n 1 presents the number of elements in A 1 n 2 represents the number of elements in A 2 and so fort nr represents the number of elements in A r, then there exist different ordered partitions of A.

12. Ex: How many distinct permutations can be formed from all the letters of each word: them ii) unusual iii) sociological i) 4! = 24, since there are 4 letters and no repitations. ii) since there are 7 letters of which 3 are u iii) since there are 12 letters of which 3 are ‘o’, 2 are ‘c’, 2 are ‘i’, 2 are ‘l’

13. b) Theorem b) the number of permutation of set A which has n elements for a circle is equal (n-1)! N people can be sit around a table in (n-1)! different form.

14. Some special combinations