2. PERMUTATIONS AND COMBINATIONS 9.1 Permutations 9.1 Permutations 9.2 Combinations

4. LEARNING OUTCOMES : At the end of the lesson students should be able to a. Understand the techniques of counting. b. Understand permutations of a set of objects. c. Find the number of permutations of n different objects. d. Find the number of permutations of r objects from n different objects.

7. Consider : 10x9x8x7x6x5x4x3x2x1 This is very long to write out so we use a shorter form, 10! read as ‘ten factorial’. Look on your calculator to find the button Check that : 5! =

8. n! = n.(n-1).(n-2).(n-3)…3.2.1, n Z + Example REMARK 9! = 9.(9-1).(9-2).(9-3)…3.2.1 = 9.(8).(7).(6).5.4.3.2.1 = 362880 0! = 1 1! = 1 n! = n.(n-1)! (n-1)! = (n-1).(n-2)!

9. The formula n! (the number of permutations of a set of n objects) has been well-known for a long time. Although it first appeared in the works of Persian and Arab mathematicians in the twelfth century, there are indications that the mathematicians of India were aware of this rule a few hundred years before Christ. However, the surprise notation ! used for “factorial” was introduced by Christian Kramp in 1808. He chose this symbol perhaps because n! get surprisingly large even for small numbers. For example, we have that 18! 6.402373x10 15, a number which, according to Karl Smith, is greater than six times “the number of all words ever printed”.

10. A permutation is an ordered arrangement of all or part of a set of objects. Given: A, B, C How many arrangements of the letters A, B, and C are there?

11. Permutation  That is six different ways

12.  The number of arrangements is 3 x 2 x 1 = 6

13. (a) The number of permutations of n different objects taken n at a time. (b) The number of permutations of n different objects taken r at a time.

14. (c) The number of permutations where some objects are repeated. (d) Conditional permutations.

15. (a) The number of permutations of n different objects taken n at a time where is

16. Example 1 In how many different ways can the letters of word STUDY be arranged? Solution: has 5 different letters which can be arranged in 5! different ways. The number of arrangements is 120

17. Example 2 How many three digit numbers can be made from the integers 3, 6, 9? Solution: The number of arrangements is 6

18. Example 3 Adi has 4 different flags. How many different signals could he make using that flags? Solution: The number of different signals is 24

19. Example 4 8 volumes of an encyclopedia are placed on a shelf. In how many different ways can it be arranged? Solution: The number of arrangements is 40320

20. Example 5 For a class photograph the five tallest students have to stand in the back row. There are three boys and two girls in the back row. How many different ways can they be arranged? Solution: The number of arrangements is 120

21. (b) The number of permutations of n different objects taken r at a time where is

22. Example 6 Find the number of permutations of the three letters A, B and C, taken two at a time. Solution:

23. Example 7 There are ten students in the class with nine chairs in one row. In how many different ways can the students sit that chairs? Solution:

24. Example 8 There are six competitors in the final of a 100m race. The first three competitors to complete the course will all receive medals. Calculate the number of different possible groups of medal winners. Solution:

25. Example 9 Adriana has eight different rose bushes, each with a different coloured flower, she will plant five of the bushes in a row in her back garden. How many different arrangement of rose bushes can she plant? Solution:

26. (a) The number of permutations of n different objects taken n at a time. (b) The number of permutations of n different objects taken r at a time.