Answer the Learning Task 1, 2, 3 on page 3.

LEARNING ACTIVITY SHEET IN MATHEMATICS 10 Quarter 3, Week 1 Learning Competency: Illustrates the permutation of objects. M10SP-illa-1 Derives the formula for finding the number of permutations of objects taken at a time. M10SP-Illa-2 Solves problems involving permutations. M10SP-IIIb-1 Illustrations of Permutations Example 1. During Fiesta, as one of our traditions, sweet delicacies are always present. Your mother prepares three types of these: Ubeng Halaya, Buko Salad, and Sweetened Macapuno. If you are supposed to help your mother in preparing the dishes to be served to your visitors, then, in how many possible ways can you serve the three sweet delicacies? Solution: By systematic listing By Tree Diagram Ubeng Halaya, Buko Salad, Macapuno Buko Salad Macapuno. Ubeng Halaya, Macapuno, Buko Salad Ubena Halayo Buko Salad, Ubeng Halaya, Macapuno Macapuno Buko Salad Buko Salad, Macapuno, Ubeng Halaya Ubena Halava Macapuno, Macspuno, Ubeng Halaya, Buko Salad Buko Salad Macapuno, Buko Salad, Ubeng Halaya Macapuno Ubena Holaya Ubena Halava Buko Salad Macapuno Buko Salad Ubena Halava As you can see from the Tree Diagram and Systematics Listing, there are 6 possible ways that you can serve sweet delicacies. FCP (Fundamental Counting Principle) If there are m ways to do one thing, n ways to do another, and o ways to do another, then, there are m x n x o of doing those things. We have : m x n x o = (3)(2)(1) = 6 possible ways of serving the sweet delicacies In this example you notice that the factors are decreasing. Another way of writing (3)(2)(1) is 3! (read as 3 factorial). Therefore, 3! = (3)(2)(1) = 6; 31 = 6 Factorial Notation If n is a positive integer, ni is a product of all positive integers less than n or equal to n. We also define 01 = 1 The Pennutation Formulas Example 2. Mother has taken fresh sitaw, lagkitang mais (white corn), saging matsing (banana). and macapuno from the farm where they lived before in some part of Brgy. Concepcion, San Pablo City. How many possible ways can we arrange the following products that are freshly taken from the farm? 1. Sitaw 2. Sitaw and Lagkitang Mais 3. Sitaw, Lagkitang Mais and Saging Matsing 4. Sitaw, Lagkitang Mais, Saging Matsing and Macapuno Solution : At this point we will use the Table to illustrate the permutations and derive the formula for finding the number of permutations. Page 1 of 5Obviously, this problem involves Circular Permutation. Thus, we are going to use the formula : P = (n -1)! There are 8 members, therefore , let's n = 8. By using the formula : P = (8-1)1 P = 7! P = (7) (6) (5) (4) (3) (2) (1) P = 5,040 ways Directions: Do the following learning tasks in a whole sheet of paper. Learning Task 1: Can you show me the way? Read the given situations and carefully answer 1. A close friend invited Anna to her birthday party. Anna has 4 new blouses (stripes, with ruffles, long-sleeved, and sleeveless) and 3 skirts (red, pink, and black) in her closet reserved for such occasions. a. Assuming that any skirt can be paired with any blouse, in how many ways can Anna select her outfit? List the possibilities. b. How many blouse-and-skirt pairs are possible? c. Show another way of finding the answer in item a. 2. How many ways are there to order the letters L,A, K,E,S? 3. In how many possible ways can you arrange 5 kilos of rambutan, 2 of kilos lanzones, 4 kilos of Indian mangoes, and 3 kilos of chicos on the table? Learning Task 2: Give Me the Number of Ways In each problem, please indicate what kind of permutations is involved, then solve. 1. How many permutations can be made to 10 pocalettes (small coconut shells) in designing the edge of a circular card? 2. The view of Sampaloc Lake is a very nice background to take pictures of. How many ways can 5 friends arrange themselves in a row for taking pictures? Learning task 3: Read the paragraph and answer the statements below. There are 7 friends who decided to go to Sampaloc Lake for a ride, however, there are only 4 available bikes. While waiting, they take a picture in a row as a souvenir. They also buy food from the stalls. They have 3 choices such as ihaw-thaw, queck-queck, and hotdog on sticks and 2 kinds of beverages (buko juice and palamig). While eating they sit on the view deck in a circular form, having so much fun. They play a game of arranging all letters found in "Sampaloc Lake" like SAMPALOC LAKE and get a new name related to it such as CLAKE, a barnacle goose. 1. Illustrate the possible ways of paring the food with beverages by using a tree diagram, systematic listing, and table. 2. In how many different ways can they arrange to give everybody a chance to experience the ride? 3. In how many ways can they arrange themselves in a row for picture taking? 4. In how many ways they can be seated on the view deck? 5. Using the letters in SAMPALOC LAKE, how many distinguishable permutations are there? Page 3 of 5. Size of Set Number of Permutation Factorial (Multiplication Rulo) Version 3 Show 2_ Show and Lookitang Mais 2(1) - 2 21 Sitow, Lagkitang Mats and (3)(2)(1) - 6 3! Saging Malsing 4 Sitaw, Lagilang Mals. (4) (3)(2)(1) - 24 Saging Matsing and Macapuno In (n-1)(n-2)(n-3)(n-4)...(3)(2)(1) | nt Therefore, the number of permutations of objects taken all at a time is n! Then, the formula for Permutations of objects taken all a time is P(n,n) = n! where n is the number of objects taken. This problem involves linear permutation. Example 3. There are 5 sweet delicacies that your mother prepared for fiesta and these were: Ubeng Halaya, Buko Salad, Sweetened Macapuno, Leche Flan, and Buko Pandan. If you are supposed to help your mother in preparing the dishes to be served to your visitors, then, in how many possible ways can you arrange the 5 delicacies if three sweet delicacies are served at a time? Let 5 = n, 3 = r Therefore, P( n,r) is the number of permutations of n objects taken r at a time. Formula : P(n, r) = 12! Solution: P (5,3) = 5! (5-3)! P(5,3) = 51 2! P (5,3) = . 5x4x3x2x1 2x1 P (5,3) = 5x4x3 P(5, 3) = 60 There are 60 ways to arrange the 5 delicacies if there are three sweet delicacies served at a time. Example 4. In how many distinguishable permutations are possible with the letters of the word PALAKPAKIN? Solution: Since the word "distinguishable" is already mentioned in the problem, obviously the formula that you are going to use is: Formula: P= p!qir! There are 10 letters in the word. 2 Ps are alike, 3 A's are alike, 2 K's are alike, therefore, we have : P =. 101 21 31 2! P - 10) (9)(8) (7)(6)(5)(4)(3)(2)(1) (2)(1)(3)(2)(1)(2)(1) P= (10)(9)(8)07)(6)(5) P = 151200 ways Example 5. There is a JHS Math Camp in the Division of San Pablo City held at the oval of Dizon High. Many students are participating from the different secondary schools. The Math Campers are grouped into 10 groups with 8 members each. Each group is asked to form a circle and they will be sitting on the ground. If the seating arrangement is circular, in how many possible ways can the 8 members be seated? Page 2 of 5