1. Combinations By Mr Porter

2. Definition: Combination Selections. The number of unordered selections of r items form n items (that is the number of combinations of r items from n items) is given by Alternate Notation: In this powerpoint, n C r, notation will be used. r - number of item you want to select. n - number of items you are selecting from.

3. Example 1 How many subgroups of 3 can be selected from a group of 10 people? The selection of the subgroup of 3 can be in any order  order is not important. Un-order Selected = 10 C 3 = 120 Use a scientific Calculator to evaluate. Students should practice the expansion simplification.

4. Example 2 A university student can borrow 4 books from the library. She wants to read 3 Biology books, 2 Mathematics, 4 Chemistry and 3 Physics books. a) How many different selections of 4 books can be made? b) How many selections can she make if she must takes only 2 Chemistry books? c) How many selections can be made if she takes at least 2 Physics books? a) Different selections of any 4 books. The total number of books = 3 + 2 + 4 + 3 =12. Selections = 12 C 4 b) Selections takes only 2 Chemistry books Select 2 of the Chemistry books first = 4 C 2 Select 2 other books from the rest of the not including the remaining Chemistry books = 8 C 2 Selections = 4 C 2 x 8 C 2 c) She takes at least 2 Physics books She can takes at 2, 3 or 4 Physics Books. 2 Physics Books = 4 C 2 3 Physics Books = 4 C 3 4 Physics Books = 4 C 4 Selections = 4 C 2 x 8 C 2 + 4 C 3 x 8 C 1 + 4 C 4 x 8 C 0

5. Example 3 A team of 3 students is to be chosen from a group of 8 girls and 5 boys to form a debating team. How many different teams can be chosen: a) without restrictions. b) if the the team is to contain a majority of girls. a) Team without restrictions Total number of students = 8 + 5 = 13 team of 3 students = 13 C 3 b) team is to contain a majority of girls There could be 2 or 3 girls in the team. 2 girls + 1 boy = 8 C 2 x 5 C 1 3 girls + 0 boys = 8 C 3 x 5 C 0 majority of girls = 8 C 2 x 5 C 1 + 8 C 3 x 5 C 0

6. Example 4 In how many ways can a committee of 4 women and 3 men be chosen from 8 women and 7 men? a) Without restrictions? b) If Sue refuses to serve if Betty is selected?. a) Without restrictions Select from each group, women then men. Committee of 4W + 3M = 8 C 4 x 7 C 3 b) Sue refuses to serve if Betty Select Betty and Sue plus 2W = 2 C 2 x 6 C 2. Select 3M remains the same = 7 C 3 Committee without BOTH Sue and Betty = unrestricted – (Betty and Sue) selected = 8 C 4 x 7 C 3 – 2 C 2 x 6 C 2 x 7 C 3

7. Example 5 In person must answer exactly 7 of 10 questions in an examination. Given that they must answer at least 3 of the first 5 questions, find the number of ways in which 7 questions can be selected. The person can select 3, 4 or all 5 from the first 5 questions and MUST choose 4, 3 and 2 from the remaining 5 questions. exactly 7 of 10 questions = 5 C 3 5 C 4 + 5 C 4 5 C 3 + 5 C 5 5 C 2

8. Example 6 Find the number of different, selections of 5 letters that can be made from the letters of the word SYLLABUS. The word SYLLABUS has 8 letters, made up of 2 ‘L’, 2’S’ and 4 non-repeating letters {Y, A, B, U}. We can select 5 letters as follows: a) 2L, 2S, 1 from {Y, A, B, U}= 2 C 2 2 C 2 4 C 1 b) 2L, 3 from {S, Y, A, B, U} = 2 C 2 5 C 3 c) 2S, 3 from {Y, L, A, B, U} = 2 C 2 5 C 3 d) 5 from {S, Y, L, A, B, U}= 6 C 5 Number of 5 letters that ‘words’= 2 C 2 2 C 2 4 C 1 + 2 C 2 5 C 3 + 2 C 2 5 C 3 + 6 C 5 Use a scientific Calculator to evaluate.

9. Example 7 Find the number of different, selections of 4 letters that can be made from the letters of the word CONCYCLIC. The word CONCYCLIC has 9 letters, made up of 4 ‘C’ and 5 non-repeating letters {O, N, Y, L, I}. We can select 4 letters as follows: a) 4 C, 0 from {O, N, Y, L, I} = 4 C 4 5 C 0 b) 3 C, 1 from {O, N, Y, L, I} = 3 C 3 5 C 1 c) 2 C, 2 from {O, N, Y, L, I} = 2 C 2 5 C 2 d) 1 C, 3 from {O, N, Y, L, I} = 1 C 1 5 C 3 e) 0 C, 4 from {O, N, Y, L, I} = 0 C 0 5 C 4 Number of 4 letters selections = 4 C 4 5 C 0 + 3 C 3 5 C 1 + 2 C 2 5 C 2 + 1 C 1 5 C 3 + 0 C 0 5 C 4 Use a scientific Calculator to evaluate.