Linear Permutation - A Grade 10 lesson under the third quarter of school year 2023-2024.

Grade 10 lesson - Linear Permutations

1. LINEAR PERMUTATION

2. LINEAR PERMUTATION - this is an ordered arrangements of objects in a line . Rule no. 1: taken all at once The number of permutations of ‘n’ distinct objects taken all at once is denoted by n P n and defined as n P n =  

3. Rule no. 1 Examples: In how many ways can 3 different books be arranged in a shelf? How many possible arrangements are there in the word MEAT? In how many possible arrangement can Anna, Jane and Mary sit on a bench?

4. Rule no. 2: taken r at a time The number of permutations of ‘n’ distinct objects taken r at a time is defined as n P r and defined as n P r =   LINEAR PERMUTATION

5. Rule no. 2 Examples: In how many ways can 3 different books be arranged in a shelf if in there are only two spaces left? In how many possible arrangement can 5 girls be seated in a 3-seater bench? In how many ways can 8 sangguniang bayan members be selected from 15 candidates?

6. Rule no. 2: Taken r at a time The number of permutations of ‘n’ distinct objects taken r at a time is defined as n P n and defined as n P r =   Rule no. 1: Taken all at once The number of permutations of ‘n’ distinct objects taken all at once is denoted by n P n and defined as n P n =   LINEAR PERMUTATION

7. Illustrative Examples Example 1 . In how many ways can 3 different books be arranged in a shelf? Step 1. Analyze the problem and i dentify what rule to use. Since we are going to arrange all the data at once we are going to use rule no. 1. n P n =   Step 2. Identify the given in the problem. In the problem, we have 3 books to arrange therefore our n=3. Step 3 . Solve the problem

8. Step 3 . Solve the problem Solution: n=3 n P n = 3 P 3 = 3 3 P 3 = 3●2 ●1 3 P 3 = 6 ways   Therefore, there are 6 different ways to arrange the 3 books.

9. Illustrative Examples Example 1 . In how many ways can 3 different books be arranged in a shelf if there are only two spaces left? Step 1. Analyze the problem and i dentify what rule to use. In the problem we are asked to arrange 3 different books. However, we can only arrange two different books at a time. There we need to use rule no. 2. n P r =   Step 2. Identify the given in the problem. In the problem, we have 3 books to arrange but we can only take 2 books at a time therefore our n=3 and r=2.

10. Step 3 . Solve the problem Solution: n = 3 and r = 2 Therefore, there are 6 ways to arrange the 3 different books if there are only two spaces left . n P r = 3 P 2 = 3 P 2 = 3 P 2 = 3 P 2 =  

11. Let’s Practice

12. Identify what is the appropriate rule to use for the following permutation problems.

13. How many ways can the letters A, B, and C be arranged? Answer: Rule no. 1: Taken all at once

14. A committee of 5 members needs to be formed from a group of 10 people. In how many ways can the committee be selected? Answer: Rule no. 2: Taken r at a time

15. In how many ways can a president, a treasurer and a secretary be chosen from among 8 candidates? Answer: Rule no. 2: Taken r at a time

16. In how many ways can 4 different books be arranged on a shelf? Answer: Rule no. 1: Taken all at once

17. In a singing competition with 10 contestants, in how many ways can the organizer arrange the first three singers? Answer: Rule no. 2: Taken r at a time

18. Solve the following problems involving permutations. How many ways can the letters A, B, and C be arranged? A committee of 5 members needs to be formed from a group of 10 people. In how many ways can the committee be selected? In how many ways can a president, a treasurer and a secretary be chosen from among 8 candidates? In how many ways can 4 different books be arranged on a shelf?

19. Assessment Sioco n National Science High School held a singing contest during the Values Month Celebration. Each grade level has one representative. In how many ways can the participants be ranked from 1 st for 6 th place? If only the top 3 finalists can receive awards, in how many different ways can 1 st , 2 nd and 3 rd be awarded be awarded?

20. c) A license plate begins with three letters. If the possible letters are A, B, C, D and E, how many different permutations of these letters can be made if no letter is used more than once? d) In how many ways can a president, a treasurer and a secretary be chosen from among 7 candidates?

21. 8 cyclists participated in the Regional Qualifying Meet. In how many ways can the participants be ranked from 1 st for 8 th place? If only the top 3 finalists can move to the Palarong Pambansa, in how many different ways can 1 st , 2 nd and 3 rd ranks be awarded? Group Activity