2. LEARNING OUTCOMES : a. understand combinations of a set of objects. b. determine the number of ways to form combinations of r objects from n objects Click for introduction

3. A) Given: Three books A, B, C How many arrangements of two books can be formed from the three books? RECALL….

4. Arrangements  That are six different ways Books

5. Given: Three books A, B, C How many combinations of two can be formed from the three books? A combination of a set of objects is a selection of objects where order does not taken into consideration.

6. Combinations  That are six different ways Books

7. BUT… AND are same combinations are same combinations AND 1+ 1 = 3 or

8. NOTES COMBINATIONS: Arrangements is NOT important PERMUTATIONS : Arrangements is important

9. The number of combinations of r objects taken from n distinct objects, without regards to the order of selection, is denoted by

10. Example 1 In how many ways could a quiz team of four be chosen from a group of fifteen students. Solution The number of ways is

11. Example 2 In how many ways could a group of 5 be chosen from a group of 3 girls and 6 boys. Solution 5 members had been chosen from a group of 9 The number of ways is

12. Example 3 The letter a, b, c, d, e, f, g, h, and i are to be divided into three groups containing two, three and four letter, respectively. In how many ways can this be done? Click me to get the solution

13. Example 4 Find the number of positive divisors of 2 x 3 x 5 x 7 x 11 Solution 2, 3, 5, 7, 11 are prime numbers Therefore, the number of divisors is the number of product that can be obtained by multiplying out some of the number 2, 3, 5, 7,11

14. The number of Prime divisors : Two-factor divisors : Three-factor divisors :

15. Four-factor divisors : Total is 5 + 10 + 10 + 5 =30 However, we have to add two more divisors, i.e. the number itself and one. Altogether we have 32 divisors.

16. Example 5 Five students were chosen from a group of eight boys and five girls. In how many ways could the group be chosen if there are to be more boys than girls in that group?

17. Boys(8)Girls(5) Solution 50 41 32 Boys(8)Girls(5) No of combinations 56 x 1 = 56 70 x 5 = 350 56 x 10 = 560 Total = 5 Boys > Girls Total of Combinations 56 + 350 + 560 = 966

18. A committee consisting of three members is to be formed from a group of 25 including five women. How many different committees can be chosen : a)without any restrictions b)if there must be at least a woman. Example 6

19. Solution a)3 persons had been chosen from 25 Man (20) Woman (5) No. of combinations TOTAL 2 30 12 1 M WTotal 950 200 10 1 160 3 20M5W b)

20. Example 7 In one of the mathematics examination, student will be given 8 questions. Students have to answer only 5 questions. Find how many ways the student may choose their questions, a)without any restrictions b)question 2 and 4 are compulsory c)If question 5 is chosen, then question 6 should be ignored.

21. a) without any restrictions Choose 5 out of 8 questions, the number of ways is Solution

22. b)question 2 and 4 are compulsory Choose Q 2 Choose Q 4 Choose other Q ( 3 out of 6 ) Total = 5 questions

23. To choose Q5 c) If question 5 is chosen, then question 6 cannot be chosen 2 cases : Q 5 is chosen Q 5 is not chosen 12 3 4567 8 12 3 4567 8 To choose another 4Q out of 6Q (Q6 not included)

24. Homework 1.A deck of alphabet cards consists of 52 cards, with 26 printed with capital letters A, B, C,…,Z and the others with lower case letters a, b, c, …, z. If you choose a collection of 13 cards from this deck, it is called a hand of 13 cards. a) How many hands of 13 cards are possible? b) How many of these hands will have: i) both the letter “A” and “a”. ii) exactly five of the ten vowel cards? (a, e, i, o, u, A, E, I, O, U) Click here for solution

25. 2. A team of 7 players is to be chosen from a group of 12 players. One of the seven is then to be elected captain and another one is to be the vice-captain. In how many ways can this be done? 3.How many are there in a class, if a two- person committee can be chosen in 300 ways? Ans: 25 ways Click here for solution