Counting

1. Counting CSCA67 Fall, 2012 Nov. 12, 2012 TA: Yadi Zhao www.utsc.utoronto.ca/~09zhaoya

2. Permutation An arrangement (or ordering) of a set of objects is called a permutation. (We can also arrange just part of the set of objects.) In a permutation, the order that we arrange the objects is important

3. Example 1 Consider arranging 3 letters: A, B, C. How many ways can this be done?

4. Theorem 1 n distinct objects can be arranged in n! ways

5. Example 2 In how many ways can 4 different students be arranged in a row?

6. Theorem 2 The number of permutations of n distinct objects taken r at a time, where repetitions are not allowed, is given by n!/(n-r)!

7. Example 3 In how many ways can a supermarket manager display 5 brands of cereals in 3 spaces on a shelf?

8. Example 4 How many different number-plates for cars can be made if each number-plate contains four of the digits 0 to 9 followed by a letter A to Z, assuming that (a) no repetition of digits is allowed? (b) repetition of digits is allowed?

9. Theorem 3 The number of different permutations of n objects of which n1are of one kind, n2are of a second kind, ... nkare of a k-th kind is n! / (n1! * n2! *…nk!)

10. Example 5 In how many ways can the six letters of the word "mammal" be arranged in a row?

11. Theorem 4 There are (n−1)! ways to arrange n distinct objects in a circle.

12. Example 6 In how many ways can 5 people be arranged in a circle?

13. Exercise 1 How many numbers greater than 1000 can be formed with the digits 3,4,6,8,9 if a digit cannot occur more than once in a number?

14. Exercise 2 How many different ways can 3 red, 4 yellow and 2 blue bulbs be arranged in a string of Christmas tree lights with 9 sockets?

15. Combination A combination of n objects taken r at a time is a selection which does not take into account the arrangement of the objects. That is, the order is not important.

16. Number of Combinations The number of ways (or combinations) in which r objects can be selected from a set of n objects, where repetition is not allowed, is denoted by: n! / (r!*(n-r)! )

17. Example 1 How many different sets of 4 different letters can be selected from the alphabet?

18. Example 2 Find the number of ways in which 3 components can be selected from a batch of 20 different components.

19. Example 3 In how many ways can a group of 4 boys be selected from 10 if (a) the eldest boy is included in each group? (b) the eldest boy is excluded? (c) What proportion of all possible groups contain the eldest boy?

20. Example 4 A class consists of 15 students of whom 5 are girls. How many committees of 8 can be formed if each consists of (a) exactly 2 girls? (b) at least 2 girls?

21. Example 5 Out of 5 mathematicians and 7 engineers, a committee consisting of 2 mathematicians and 3 engineers is to be formed. In how many ways can this be done if (a) any mathematician and any engineer can be included?

22. Example 5 Out of 5 mathematicians and 7 engineers, a committee consisting of 2 mathematicians and 3 engineers is to be formed. In how many ways can this be done if (b) one particular engineer must be in the committee?

23. Example 5 Out of 5 mathematicians and 7 engineers, a committee consisting of 2 mathematicians and 3 engineers is to be formed. In how many ways can this be done if (c) two particular mathematicians cannot be in the committee?